This dataset contains all 238,764,310 elliptic curves over $\Q$ with naive height up to $2.7 \cdot 10^{10}$ and is described in Balakrishnan-Ho-Kaplan-Spicer-Stein-Watkins.
File and data format
The data is stored in text files, indexed from $k=0$ to $k=2699$. The file $k$.txt contains data for elliptic curves $y^2 = x^3 + A x + B$ with $k \cdot 10^7 < H \le (k+1) \cdot 10^7$, where $H:=\max(4|A|^3,27B^2)$.
Each line corresponds to an elliptic curve, with columns separated by | characters and format given as follows. If a column has not been computed it is saved as the empty string.
| Column(s) | Description | Example |
|---|---|---|
| $H$ | Naive height | $19003704300$ |
| $a_1$, $a_2$, $a_3$, $a_4$, $a_6$ | $a$-invariants for the reduced minimal model | $1$, $-1$, $0$, $-102$, $-389$ |
| $p_1$, $p_2$ | Parameters to form the elliptic curve, which is isomorphic to $y^2 = x^3 + p_1 x + p_2$ | $-1635$, $-26530$ |
| $N$ | Conductor | $5940675$ |
| $\operatorname{tam}$ | Tamagawa product | $1$ |
| $n_1$, $n_2$ | structure of torsion subgroup, $\Z/n_1 \times \Z/n_2$ where $n_1 \in {1, 2}$ and $n_1 \mid n_2$ | $1$, $1$ |
| $\operatorname{sel}_2$ | $2$-rank of the $2$-Selmer group | $2$ |
| $w$ | Root number (sign of functional equation) | $-1$ |
| $r_{\mathrm{an},0}$ | Running analytic_rank_upper_bound in Sage with parameter $\delta$ (see below) returns $r_{\mathrm{an},0}$, which is an upper bound for the rank | $0$ |
| $\operatorname{mw}_{\mathrm{ub}}, \operatorname{mw}_{\mathrm{lb}}, \operatorname{mw}_{\mathrm{time}}$ | Upper and lower bounds returned by mwrank in Sage, as well as the the time used | $2, 2, 0.052$ |
| $\delta$ | Running analytic_rank_upper_bound in Sage with parameter $\delta$ returns $r_{\mathrm{an},0}$ (see above), which is an upper bound for the rank. | $2.0$ |
| $\operatorname{magma}$ | Rank computed by the Magma function MordellWeilShaInformation (almost always null) | $2$ |
| $r$ | rank | $4$ |
| $\operatorname{CM}$ | 1 if $E$ has CM, 0 otherwise | $0$ |