| 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 |
| 108.1-a1 |
108.1-a |
$8$ |
$20$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
108.1 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{40} \cdot 3^{8} \) |
$1.03864$ |
$(-a), (-a+1), (2)$ |
$0$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2^{4} \cdot 5 \) |
$1$ |
$0.802078487$ |
1.112282735 |
\( -\frac{4395631034341}{3145728} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( 1707 a - 4095\) , \( 53306 a - 122129\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(1707a-4095\right){x}+53306a-122129$ |
| 108.1-a2 |
108.1-a |
$8$ |
$20$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
108.1 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{8} \cdot 3^{16} \) |
$1.03864$ |
$(-a), (-a+1), (2)$ |
$0$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2^{4} \) |
$1$ |
$4.010392437$ |
1.112282735 |
\( \frac{5735339}{3888} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( -18 a + 45\) , \( -19 a + 46\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-18a+45\right){x}-19a+46$ |
| 108.1-a3 |
108.1-a |
$8$ |
$20$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
108.1 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{4} \cdot 3^{26} \) |
$1.03864$ |
$(-a), (-a+1), (2)$ |
$0$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2Cs, 5B.4.1 |
$1$ |
\( 2^{4} \) |
$1$ |
$4.010392437$ |
1.112282735 |
\( \frac{476379541}{236196} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( 82 a - 195\) , \( -239 a + 538\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(82a-195\right){x}-239a+538$ |
| 108.1-a4 |
108.1-a |
$8$ |
$20$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
108.1 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{2} \cdot 3^{31} \) |
$1.03864$ |
$(-a), (-a+1), (2)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$4.010392437$ |
1.112282735 |
\( -\frac{1025795879759761}{3486784401} a + \frac{5304841542920801}{6973568802} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( 1072 a - 2535\) , \( -26573 a + 61324\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(1072a-2535\right){x}-26573a+61324$ |
| 108.1-a5 |
108.1-a |
$8$ |
$20$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
108.1 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{2} \cdot 3^{31} \) |
$1.03864$ |
$(-a), (-a+1), (2)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B.4.1 |
$4$ |
\( 2^{2} \) |
$1$ |
$1.002598109$ |
1.112282735 |
\( \frac{1025795879759761}{3486784401} a + \frac{1084416594467093}{2324522934} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( 692 a - 1695\) , \( 13615 a - 31880\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(692a-1695\right){x}+13615a-31880$ |
| 108.1-a6 |
108.1-a |
$8$ |
$20$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
108.1 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{10} \cdot 3^{11} \) |
$1.03864$ |
$(-a), (-a+1), (2)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B.4.2 |
$4$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$0.200519621$ |
1.112282735 |
\( -\frac{1373276865151726904870180471}{1296} a + \frac{2108232339288241560240379517}{864} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( -83860 a - 122846\) , \( 19206619 a + 24409255\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-83860a-122846\right){x}+19206619a+24409255$ |
| 108.1-a7 |
108.1-a |
$8$ |
$20$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
108.1 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{20} \cdot 3^{10} \) |
$1.03864$ |
$(-a), (-a+1), (2)$ |
$0$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2Cs, 5B.4.2 |
$1$ |
\( 2^{4} \cdot 5 \) |
$1$ |
$0.802078487$ |
1.112282735 |
\( \frac{18013780041269221}{9216} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( 27307 a - 65535\) , \( 3437626 a - 7866641\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(27307a-65535\right){x}+3437626a-7866641$ |
| 108.1-a8 |
108.1-a |
$8$ |
$20$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
108.1 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{10} \cdot 3^{11} \) |
$1.03864$ |
$(-a), (-a+1), (2)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$0.802078487$ |
1.112282735 |
\( \frac{1373276865151726904870180471}{1296} a + \frac{3578143287561270870980777609}{2592} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( 16267 a - 79935\) , \( 4343002 a - 6682673\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(16267a-79935\right){x}+4343002a-6682673$ |
| 108.1-b1 |
108.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
108.1 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{4} \cdot 3^{6} \) |
$1.03864$ |
$(-a), (-a+1), (2)$ |
$0$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$16.24908073$ |
1.502228045 |
\( -\frac{344113}{108} a - \frac{149305}{36} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -a + 1\) , \( a - 5\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-a+1\right){x}+a-5$ |
| 108.1-b2 |
108.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
108.1 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{12} \cdot 3^{10} \) |
$1.03864$ |
$(-a), (-a+1), (2)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$1.805453415$ |
1.502228045 |
\( \frac{4653908}{3} a - \frac{228516739}{64} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 90 a + 117\) , \( 117 a + 152\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(90a+117\right){x}+117a+152$ |
| 108.1-b3 |
108.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
108.1 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{6} \cdot 3^{11} \) |
$1.03864$ |
$(-a), (-a+1), (2)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$1.805453415$ |
1.502228045 |
\( -\frac{312258622128767}{36} a + \frac{479374365181397}{24} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( -350 a - 483\) , \( -499 a - 688\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-350a-483\right){x}-499a-688$ |
| 108.1-b4 |
108.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
108.1 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{2} \cdot 3^{9} \) |
$1.03864$ |
$(-a), (-a+1), (2)$ |
$0$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$16.24908073$ |
1.502228045 |
\( \frac{60298633043}{1458} a + \frac{13100436649}{243} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( 9 a - 29\) , \( 21 a - 47\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(9a-29\right){x}+21a-47$ |
| 108.1-c1 |
108.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
108.1 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{4} \cdot 3^{12} \) |
$1.03864$ |
$(-a), (-a+1), (2)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.131974785$ |
0.868653514 |
\( -\frac{344113}{108} a - \frac{149305}{36} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -a\) , \( 3 a - 8\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}-a{x}+3a-8$ |
| 108.1-c2 |
108.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
108.1 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{12} \cdot 3^{4} \) |
$1.03864$ |
$(-a), (-a+1), (2)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.131974785$ |
0.868653514 |
\( \frac{4653908}{3} a - \frac{228516739}{64} \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( 18 a + 22\) , \( -17 a - 23\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(18a+22\right){x}-17a-23$ |
| 108.1-c3 |
108.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
108.1 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{6} \cdot 3^{5} \) |
$1.03864$ |
$(-a), (-a+1), (2)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.131974785$ |
0.868653514 |
\( -\frac{312258622128767}{36} a + \frac{479374365181397}{24} \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( -62 a - 98\) , \( 79 a + 73\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-62a-98\right){x}+79a+73$ |
| 108.1-c4 |
108.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
108.1 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{2} \cdot 3^{15} \) |
$1.03864$ |
$(-a), (-a+1), (2)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.131974785$ |
0.868653514 |
\( \frac{60298633043}{1458} a + \frac{13100436649}{243} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 9 a - 60\) , \( 41 a - 182\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(9a-60\right){x}+41a-182$ |
*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.
