| 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-a4 |
4.1-a |
$6$ |
$45$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{2} \) |
$0.45564$ |
$(2)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 5$ |
3B, 5B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$28.55651349$ |
0.316806072 |
\( \frac{461373}{2} a - 531398 \) |
\( \bigl[1\) , \( 1\) , \( a\) , \( a - 3\) , \( -a + 1\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(a-3\right){x}-a+1$ |
| 36.2-a4 |
36.2-a |
$6$ |
$45$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
36.2 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{2} \cdot 3^{6} \) |
$0.78920$ |
$(-a), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B.1.1, 5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$34.19516291$ |
1.053781309 |
\( \frac{461373}{2} a - 531398 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 2 a - 5\) , \( -5 a + 11\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(2a-5\right){x}-5a+11$ |
| 36.3-a4 |
36.3-a |
$6$ |
$45$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
36.3 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{2} \cdot 3^{6} \) |
$0.78920$ |
$(-a+1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B.1.2, 5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$3.799462546$ |
1.053781309 |
\( \frac{461373}{2} a - 531398 \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( -4 a - 4\) , \( -15 a - 20\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-4a-4\right){x}-15a-20$ |
| 256.1-c4 |
256.1-c |
$6$ |
$45$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{26} \) |
$1.28875$ |
$(2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B, 5B.4.1 |
$1$ |
\( 2 \) |
$1$ |
$3.412266367$ |
1.892784823 |
\( \frac{461373}{2} a - 531398 \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 24 a - 56\) , \( 96 a - 220\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(24a-56\right){x}+96a-220$ |
| 256.1-d4 |
256.1-d |
$6$ |
$45$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{26} \) |
$1.28875$ |
$(2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B, 5B.4.1 |
$1$ |
\( 2^{2} \) |
$0.056267535$ |
$14.80693988$ |
1.848593902 |
\( \frac{461373}{2} a - 531398 \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -8\) , \( 8 a + 20\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}-8{x}+8a+20$ |
| 256.1-f4 |
256.1-f |
$6$ |
$45$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{26} \) |
$1.28875$ |
$(2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B, 5B.4.1 |
$1$ |
\( 2^{2} \) |
$0.506407817$ |
$1.645215542$ |
1.848593902 |
\( \frac{461373}{2} a - 531398 \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -8\) , \( -8 a - 20\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}-8{x}-8a-20$ |
| 324.1-a4 |
324.1-a |
$6$ |
$45$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{2} \cdot 3^{12} \) |
$1.36693$ |
$(-a), (-a+1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B, 5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$4.549688489$ |
1.261856548 |
\( \frac{461373}{2} a - 531398 \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( 13 a - 32\) , \( 33 a - 78\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(13a-32\right){x}+33a-78$ |
| 676.1-d4 |
676.1-d |
$6$ |
$45$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{2} \cdot 13^{6} \) |
$1.64285$ |
$(-2a+1), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B, 5B.4.1 |
$1$ |
\( 2 \) |
$0.461048252$ |
$3.785569646$ |
1.936270087 |
\( \frac{461373}{2} a - 531398 \) |
\( \bigl[a\) , \( -a - 1\) , \( 1\) , \( -a - 7\) , \( 14 a + 10\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-a-7\right){x}+14a+10$ |
| 1156.2-b4 |
1156.2-b |
$6$ |
$45$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1156.2 |
\( 2^{2} \cdot 17^{2} \) |
\( 2^{2} \cdot 17^{6} \) |
$1.87867$ |
$(a-5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B, 5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$3.310384624$ |
0.918135500 |
\( \frac{461373}{2} a - 531398 \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( -16 a - 21\) , \( 159 a + 205\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-16a-21\right){x}+159a+205$ |
| 1156.3-b4 |
1156.3-b |
$6$ |
$45$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1156.3 |
\( 2^{2} \cdot 17^{2} \) |
\( 2^{2} \cdot 17^{6} \) |
$1.87867$ |
$(a+4), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B, 5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$3.310384624$ |
0.918135500 |
\( \frac{461373}{2} a - 531398 \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( 11 a - 25\) , \( 23 a - 50\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(11a-25\right){x}+23a-50$ |
*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.
