An elliptic curve is defined by numbers and . The curve is the set of all points of the real plane that satisfy the Equation 1. "Definition of the elliptic curves"
Equation 1.
Definition of the elliptic curves
. 
Equation 1. "Definition of the elliptic curves" definies elliptic curves over any field, it doesn't have to the real numbers. Notably, the definition also works for finite fields, leading to elliptic curve over a finite fields, which are the ones used in Elliptic-curve Diffie-Hellman cyprotgraphy.
Some of the most remarkable ones seem to be:
- Alain Aspect 1982
- Hensen et al., Giustina et al., Shalm et al. (2015): "loophole-free" Bell tests
The elliptic curve group of an elliptic curve is a group in which the elements of the group are points on an elliptic curve.
The group operation is called elliptic curve point addition.
How to use an Oxford Nanopore MinION to extract DNA from river water and determine which bacteria live in it PCR verification with gel electrophoresis by 
Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16For this reason, it is wise to verify that certain steps are correct whenever possible.
Gel electrophoresis separates molecules by their charge-to-mass ratio. It is one of those ultra common lab procedures!
Since we know that we amplified the 16S regions which we know the rough size of (there might be a bit of variability across species, but not that much), we were expecting to see a big band at that size.
And that is exactly what we saw!
First we had to prepare the gel, put the gel comb, and pipette the samples into wells present in the gel:
To see the DNA, we added ethidium bromide to the samples, which is a substance that that both binds to DNA and is fluorescent.
Because it interacts heavily with DNA, ethidium bromide is a mutagen, and the biology people sure did treat the dedicated electrophoresis bench area with respect! Figure 4. "Gel electrophoresis dedicated bench area to prevent ethidium bromide contamination.".
Gel electrophoresis dedicated waste bin for centrifuge tubes and pipette tips contaminated with ethidium bromide.
Source. The UV transilluminator we used to shoot UV light into the gel was the Fisher Scientific UVP LM-26E Benchtop 2UV Transilluminator. The fluorescent substance then emitted a light we can see.
As barely seen at Figure 8. "Fischer Scientific UVP LM-26E Benchtop 2UV Transilluminator illuminated gel." due to bad photo quality due to lack of light, there is one strong green line, which compared to the ladder matches our expected 16S length. What we saw it with the naked eyes was very clear however.
Not every belongs to the elliptic curve over a non quadratically closed field by Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16One major difference between the elliptic curve over a finite field or the elliptic curve over the rational numbers the elliptic curve over the real numbers is that not every possible generates a member of the curve.
This is because on the Equation "Definition of the elliptic curves" we see that given an , we calculate , which always produces an element .
But then we are not necessarily able to find an for the , because not all fields are not quadratically closed fields.
For example: with and , taking gives:and therefore there is no that satisfies the equation. So is not on the curve if we consider this elliptic curve over the rational numbers.
That would also not belong to Elliptic curve over the finite field , because doing everything we have:Therefore, there is no element such that or , i.e. and don't have a multiplicative inverse.
For the real numbers, it would work however, because the real numbers are a quadratically closed field, and .
For this reason, it is not necessarily trivial to determine the number of elements of an elliptic curve.
www.youtube.com/playlist?list=PLVV0r6CmEsFw1phnddYWXtVkRW8eUVlqx Edward Teller interview by Web of Stories (1996) Date shown at: www.webofstories.com/play/edward.teller/1. Listener: John H. Nuckolls
Edward Teller, An Early Time
. Source. Comissioned by the Los Alamos National Laboratory in 1979. Producer: Mario Balibreraa.- youtu.be/Goim-4MF_uE?t=338; holy fuck he almost cut his foot off on a stupid tram accident!
- youtu.be/Goim-4MF_uE?t=457: he plays the piano
- youtu.be/Goim-4MF_uE?t=965: he drove Szilard to propose to Einstein the Einstein-Szilard letter
Largest known ranks of an elliptic curve over the rational numbers by Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16web.math.pmf.unizg.hr/~duje/tors/rankhist.html gives a list with Elkies (2006) on top with:TODO why this non standard formulation?
Reduction of an elliptic curve over the rational numbers to an elliptic curve over a finite field mod p by Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16This construction takes as input:and it produces an elliptic curve over a finite field of order as output.
The constructions is used in the Birch and Swinnerton-Dyer conjecture.
To do it, we just convert the coefficients and from the Equation "Definition of the elliptic curves" from rational numbers to elements of the finite field.
For the denominator , we just use the multiplicative inverse, e.g. supposing we havewhere because , related: math.stackexchange.com/questions/1204034/elliptic-curve-reduction-modulo-p
The paper that states the BSD conjecture.
Likely paywalled at: www.degruyter.com/document/doi/10.1515/crll.1965.218.79/html. One illegal upload at: virtualmath1.stanford.edu/~conrad/BSDseminar/refs/BSDorigin.pdf.
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
. Source. We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 3. Visual Studio Code extension installation.
Figure 4. Visual Studio Code extension tree navigation.
Figure 5. Web editor. You can also edit articles on the Web editor without installing anything locally.Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.Video 4. OurBigBook Visual Studio Code extension editing and navigation demo. Source. 
- Infinitely deep tables of contents:
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact