| 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 |
| 4.1-a3 |
4.1-a |
$6$ |
$45$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{30} \) |
$0.45564$ |
$(2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 5$ |
3Cs, 5B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$1.142260539$ |
0.316806072 |
\( -\frac{1680914269}{32768} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( 75 a - 172\) , \( 507 a - 1170\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(75a-172\right){x}+507a-1170$ |
| 36.2-a3 |
36.2-a |
$6$ |
$45$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
36.2 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{30} \cdot 3^{6} \) |
$0.78920$ |
$(-a), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3Cs.1.1, 5B.4.2 |
$1$ |
\( 3 \cdot 5 \) |
$1$ |
$2.279677527$ |
1.053781309 |
\( -\frac{1680914269}{32768} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 124 a - 297\) , \( 1087 a - 2492\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(124a-297\right){x}+1087a-2492$ |
| 36.3-a3 |
36.3-a |
$6$ |
$45$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
36.3 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{30} \cdot 3^{6} \) |
$0.78920$ |
$(-a+1), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3Cs.1.1, 5B.4.2 |
$1$ |
\( 3 \cdot 5 \) |
$1$ |
$2.279677527$ |
1.053781309 |
\( -\frac{1680914269}{32768} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -124 a - 173\) , \( -1087 a - 1405\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-124a-173\right){x}-1087a-1405$ |
| 256.1-c3 |
256.1-c |
$6$ |
$45$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{54} \) |
$1.28875$ |
$(2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3Cs, 5B.4.2 |
$1$ |
\( 2 \) |
$1$ |
$3.412266367$ |
1.892784823 |
\( -\frac{1680914269}{32768} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 1189 a - 2774\) , \( -30433 a + 70154\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(1189a-2774\right){x}-30433a+70154$ |
| 256.1-d3 |
256.1-d |
$6$ |
$45$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{54} \) |
$1.28875$ |
$(2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3Cs, 5B.4.2 |
$1$ |
\( 2^{2} \) |
$0.844013029$ |
$0.987129325$ |
1.848593902 |
\( -\frac{1680914269}{32768} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( a - 395\) , \( 1581 a - 988\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(a-395\right){x}+1581a-988$ |
| 256.1-f3 |
256.1-f |
$6$ |
$45$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{54} \) |
$1.28875$ |
$(2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3Cs, 5B.4.2 |
$1$ |
\( 2^{2} \) |
$0.844013029$ |
$0.987129325$ |
1.848593902 |
\( -\frac{1680914269}{32768} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( a - 395\) , \( -1581 a + 988\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(a-395\right){x}-1581a+988$ |
| 324.1-a3 |
324.1-a |
$6$ |
$45$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{30} \cdot 3^{12} \) |
$1.36693$ |
$(-a), (-a+1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3Cs, 5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$4.549688489$ |
1.261856548 |
\( -\frac{1680914269}{32768} \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 669 a - 1560\) , \( -13229 a + 30488\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(669a-1560\right){x}-13229a+30488$ |
| 676.1-d3 |
676.1-d |
$6$ |
$45$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{30} \cdot 13^{6} \) |
$1.64285$ |
$(-2a+1), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3, 5$ |
3Cs, 5B.4.2 |
$1$ |
\( 2 \cdot 3 \cdot 5 \) |
$0.030736550$ |
$3.785569646$ |
1.936270087 |
\( -\frac{1680914269}{32768} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -322\) , \( 2127\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-322{x}+2127$ |
| 1156.2-b3 |
1156.2-b |
$6$ |
$45$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1156.2 |
\( 2^{2} \cdot 17^{2} \) |
\( 2^{30} \cdot 17^{6} \) |
$1.87867$ |
$(a-5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3Cs, 5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$3.310384624$ |
0.918135500 |
\( -\frac{1680914269}{32768} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 1\) , \( -521 a - 767\) , \( 9127 a + 12156\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-521a-767\right){x}+9127a+12156$ |
| 1156.3-b3 |
1156.3-b |
$6$ |
$45$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1156.3 |
\( 2^{2} \cdot 17^{2} \) |
\( 2^{30} \cdot 17^{6} \) |
$1.87867$ |
$(a+4), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3Cs, 5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$3.310384624$ |
0.918135500 |
\( -\frac{1680914269}{32768} \) |
\( \bigl[a\) , \( 1\) , \( 1\) , \( 520 a - 1287\) , \( -9127 a + 21283\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+{x}^{2}+\left(520a-1287\right){x}-9127a+21283$ |
*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.
