This dataset contains 136,924,520 elliptic curves over $\Q$ with conductor up to $10^8$, divided into 115,821,258 isogeny classes. It also contains 11,378,911 elliptic curves with prime conductor up to $10^{10}$, divided into 11,372,286 isogeny classes. Not every curve satisfying these conductor bounds is present in this dataset, only those that satisfy additional bounds on the discriminant and coefficients are included; see Stein-Watkins for details.
File and data format
The data for conductors $N\le 10^8$ is stored in 1000 text files, indexed from $k=0$ to $k=999$, where the $k$th file contains data for $k \cdot 10^5 < N \le (k+1) \cdot 10^5$. Similarly, the data for prime conductors $p\le 10^{10}$ is stored in 100 text files, indexed from $k=0$ to $k=99$, where the $k$th file contains data for $k \cdot 10^8 < p \le (k+1) \cdot 10^8$.
Within each file, curves $E$ are grouped into isogeny classes. The first row of an isogeny class has the following form, with columns separated by spaces. In general, X is used when a quantity has not been computed.
| Column | Description | Example |
|---|---|---|
| $N$ | The conductor of $E$ | $2200005$ |
| $[p_1,\dots,p_m]$ | The primes dividing the conductor in increasing order | $[3,5,48889]$ |
| $r$ | The rank of $E$ | $2$ |
| $L^{(r)}(1)/r!$ | The special value of $E$ | $14.921134$ |
| $\operatorname{isog}_{\mathrm{max}}$ | The largest degree of an isogeny between non-isomorphic curves in the class | $12$ |
| $\operatorname{deg}_{\mathrm{mod}}$ | The modular degree of the optimal quotient of $J_0(N)$. A star indicates that the curve of minimal Faltings height is not $J_0$-optimal.A plus indicates that the minimal Faltings height curve is a minimal quadratic twist. |
+*2 |
Later rows in an isogeny class correspond to individual curves within that class (these rows are distinguished by starting with a square bracket rather than an integer).
| Column | Description | Example |
|---|---|---|
| $[a_1,a_2,a_3,a_4,a_6]$ | The $a$-invariants of the reduced minimal model | $[1,0,1,-1065042,-440575900]$ |
| $\operatorname{ord}(\Delta)$ | The valuations of the discriminant of $E$ at each prime dividing the conductor, in order. Surrounded by square brackets if $\Delta > 0$ and round brackets if $\Delta < 0$. |
$(12,9,2,1)$ |
| $Ш_{\mathrm{an}}$ | Analytic order of Ш | $4$ |
| $E_{\mathrm{tors}}$ | The torsion subgroup, where $n$ indicates $\Z/n\Z$ and $nx$ indicates $\Z/n\Z \times \Z/2\Z$ | $6x$ |