| Label |
Class |
Class size |
Class degree |
Base field |
Field degree |
Field signature |
Conductor |
Conductor norm |
Discriminant norm |
Root analytic conductor |
Bad primes |
Rank |
Torsion |
CM |
CM |
Sato-Tate |
$\Q$-curve |
Base change |
Semistable |
Potentially good |
Nonmax $\ell$ |
mod-$\ell$ images |
$Ш_{\textrm{an}}$ |
Tamagawa |
Regulator |
Period |
Leading coeff |
j-invariant |
Weierstrass coefficients |
Weierstrass equation |
| 8.1-a1 |
8.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{16} \) |
$1.16409$ |
$(2,a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$2.176177662$ |
$21.87465325$ |
1.536384471 |
\( -432 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -8 a - 26\) , \( -64 a - 250\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-8a-26\right){x}-64a-250$ |
| 8.1-a2 |
8.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{22} \) |
$1.16409$ |
$(2,a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$2.176177662$ |
$21.87465325$ |
1.536384471 |
\( -128955875202 a + 499443959016 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -208 a - 806\) , \( -1388 a - 5366\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-208a-806\right){x}-1388a-5366$ |
| 8.1-a3 |
8.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{20} \) |
$1.16409$ |
$(2,a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1.088088831$ |
$21.87465325$ |
1.536384471 |
\( 1000188 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -168 a - 646\) , \( -2548 a - 9870\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-168a-646\right){x}-2548a-9870$ |
| 8.1-a4 |
8.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{22} \) |
$1.16409$ |
$(2,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$2.176177662$ |
$5.468663313$ |
1.536384471 |
\( 128955875202 a + 499443959016 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -2688 a - 10406\) , \( -152764 a - 591654\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-2688a-10406\right){x}-152764a-591654$ |
| 8.1-b1 |
8.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{4} \) |
$1.16409$ |
$(2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$21.87465325$ |
1.412002795 |
\( -432 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -125 a - 476\) , \( 3263 a + 12642\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-125a-476\right){x}+3263a+12642$ |
| 8.1-b2 |
8.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{10} \) |
$1.16409$ |
$(2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$5.468663313$ |
1.412002795 |
\( -128955875202 a + 499443959016 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -3235 a - 12521\) , \( 67026 a + 259595\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-3235a-12521\right){x}+67026a+259595$ |
| 8.1-b3 |
8.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{8} \) |
$1.16409$ |
$(2,a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$4$ |
\( 2 \) |
$1$ |
$21.87465325$ |
1.412002795 |
\( 1000188 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -2605 a - 10081\) , \( 141300 a + 547257\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-2605a-10081\right){x}+141300a+547257$ |
| 8.1-b4 |
8.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{10} \) |
$1.16409$ |
$(2,a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$21.87465325$ |
1.412002795 |
\( 128955875202 a + 499443959016 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -41655 a - 161321\) , \( 9092142 a + 35213719\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-41655a-161321\right){x}+9092142a+35213719$ |
| 8.1-c1 |
8.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{16} \) |
$1.16409$ |
$(2,a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$2.176177662$ |
$21.87465325$ |
1.536384471 |
\( -432 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -8 a - 26\) , \( 64 a + 250\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-8a-26\right){x}+64a+250$ |
| 8.1-c2 |
8.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{22} \) |
$1.16409$ |
$(2,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$2.176177662$ |
$5.468663313$ |
1.536384471 |
\( -128955875202 a + 499443959016 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -208 a - 806\) , \( 1388 a + 5366\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-208a-806\right){x}+1388a+5366$ |
| 8.1-c3 |
8.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{20} \) |
$1.16409$ |
$(2,a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1.088088831$ |
$21.87465325$ |
1.536384471 |
\( 1000188 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -168 a - 646\) , \( 2548 a + 9870\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-168a-646\right){x}+2548a+9870$ |
| 8.1-c4 |
8.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{22} \) |
$1.16409$ |
$(2,a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$2.176177662$ |
$21.87465325$ |
1.536384471 |
\( 128955875202 a + 499443959016 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -2688 a - 10406\) , \( 152764 a + 591654\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-2688a-10406\right){x}+152764a+591654$ |
| 8.1-d1 |
8.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{4} \) |
$1.16409$ |
$(2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$21.87465325$ |
1.412002795 |
\( -432 \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -4\) , \( -2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}-4{x}-2$ |
| 8.1-d2 |
8.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{10} \) |
$1.16409$ |
$(2,a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$21.87465325$ |
1.412002795 |
\( -128955875202 a + 499443959016 \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( 10 a - 49\) , \( -36 a + 125\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(10a-49\right){x}-36a+125$ |
| 8.1-d3 |
8.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{8} \) |
$1.16409$ |
$(2,a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$4$ |
\( 2 \) |
$1$ |
$21.87465325$ |
1.412002795 |
\( 1000188 \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -9\) , \( -2 a - 7\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}-9{x}-2a-7$ |
| 8.1-d4 |
8.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{10} \) |
$1.16409$ |
$(2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$5.468663313$ |
1.412002795 |
\( 128955875202 a + 499443959016 \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -10 a - 49\) , \( -56 a - 219\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-10a-49\right){x}-56a-219$ |
*The rank, regulator and analytic order of Ш are
not known for all curves in the database; curves for which these are
unknown will not appear in searches specifying one of these
quantities.
