| 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 |
| 100.1-e2 |
100.1-e |
$2$ |
$13$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
100.1 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{26} \cdot 5^{2} \) |
$1.16510$ |
$(-a+2), (-a-1), (5)$ |
0 |
$\Z/13\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$13$ |
13B.1.1 |
$1$ |
\( 13^{2} \) |
$1$ |
$11.18083146$ |
2.711749948 |
\( -\frac{60698457}{40960} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -131 a - 205\) , \( 1758 a + 2745\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-131a-205\right){x}+1758a+2745$ |
| 27.1-a2 |
27.1-a |
$2$ |
$13$ |
\(\Q(\zeta_{7})^+\) |
$3$ |
$[3, 0]$ |
27.1 |
\( 3^{3} \) |
\( - 3^{3} \) |
$1.08342$ |
$(3)$ |
0 |
$\Z/13\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$13$ |
13B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$476.2115463$ |
0.402545685 |
\( -\frac{28672}{3} \) |
\( \bigl[0\) , \( -a\) , \( a\) , \( 2 a^{2} - a - 4\) , \( -2 a^{2} + a + 4\bigr] \) |
${y}^2+a{y}={x}^{3}-a{x}^{2}+\left(2a^{2}-a-4\right){x}-2a^{2}+a+4$ |
| 25.1-a2 |
25.1-a |
$2$ |
$13$ |
4.4.725.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{2} \) |
$3.59791$ |
$(-2a^3+2a^2+4a-1)$ |
0 |
$\Z/13\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$13$ |
13B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$2584.589597$ |
0.567983715 |
\( \frac{192512}{5} a^{3} - \frac{192512}{5} a^{2} - \frac{385024}{5} a - \frac{118784}{5} \) |
\( \bigl[0\) , \( -a + 1\) , \( a^{2} - a - 1\) , \( a^{3} - a^{2} - 3 a + 2\) , \( 4 a^{3} - 6 a^{2} - 9 a + 8\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(a^{3}-a^{2}-3a+2\right){x}+4a^{3}-6a^{2}-9a+8$ |
| 4.2-b1 |
4.2-b |
$2$ |
$13$ |
4.4.16448.2 |
$4$ |
$[4, 0]$ |
4.2 |
\( 2^{2} \) |
\( 2^{26} \) |
$13.62864$ |
$(-2a^2+3a+8), (-2a^3-2a^2+9a+11)$ |
0 |
$\Z/13\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$13$ |
13B.1.1 |
$1$ |
\( 13^{2} \) |
$1$ |
$257.7913978$ |
2.010073198 |
\( \frac{49375}{128} a^{2} - \frac{49375}{128} a - \frac{63875}{64} \) |
\( \bigl[a^{3} - 5 a - 3\) , \( a\) , \( a^{3} - 4 a - 4\) , \( 9 a^{3} + 7 a^{2} - 43 a - 45\) , \( -132 a^{3} - 14 a^{2} + 906 a + 873\bigr] \) |
${y}^2+\left(a^{3}-5a-3\right){x}{y}+\left(a^{3}-4a-4\right){y}={x}^{3}+a{x}^{2}+\left(9a^{3}+7a^{2}-43a-45\right){x}-132a^{3}-14a^{2}+906a+873$ |
| 64.1-a2 |
64.1-a |
$2$ |
$13$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
64.1 |
\( 2^{6} \) |
\( 2^{6} \) |
$69.23164$ |
$(2)$ |
0 |
$\Z/13\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$13$ |
13B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$123898.9272$ |
1.33823 |
\( -\frac{189}{2} \) |
\( \bigl[-2 a^{5} + a^{4} + 14 a^{3} + 4 a^{2} - 9 a - 2\) , \( -3 a^{5} + a^{4} + 21 a^{3} + 8 a^{2} - 10 a - 1\) , \( -6 a^{5} + 2 a^{4} + 43 a^{3} + 17 a^{2} - 29 a - 6\) , \( -6 a^{5} + a^{4} + 43 a^{3} + 24 a^{2} - 24 a - 7\) , \( -10 a^{5} + 2 a^{4} + 72 a^{3} + 37 a^{2} - 42 a - 14\bigr] \) |
${y}^2+\left(-2a^{5}+a^{4}+14a^{3}+4a^{2}-9a-2\right){x}{y}+\left(-6a^{5}+2a^{4}+43a^{3}+17a^{2}-29a-6\right){y}={x}^{3}+\left(-3a^{5}+a^{4}+21a^{3}+8a^{2}-10a-1\right){x}^{2}+\left(-6a^{5}+a^{4}+43a^{3}+24a^{2}-24a-7\right){x}-10a^{5}+2a^{4}+72a^{3}+37a^{2}-42a-14$ |
| 729.1-a2 |
729.1-a |
$2$ |
$13$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
729.1 |
\( 3^{6} \) |
\( 3^{6} \) |
$84.79110$ |
$(3)$ |
$1$ |
$\Z/13\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$13$ |
13B.1.1 |
$1$ |
\( 1 \) |
$0.258046207$ |
$226777.4369$ |
3.79237 |
\( -\frac{28672}{3} \) |
\( \bigl[0\) , \( -2 a^{5} + 15 a^{3} + 10 a^{2} - 10 a - 5\) , \( -2 a^{5} + 15 a^{3} + 10 a^{2} - 10 a - 5\) , \( 8 a^{5} - 2 a^{4} - 57 a^{3} - 28 a^{2} + 34 a + 11\) , \( -8 a^{5} + 2 a^{4} + 57 a^{3} + 28 a^{2} - 34 a - 11\bigr] \) |
${y}^2+\left(-2a^{5}+15a^{3}+10a^{2}-10a-5\right){y}={x}^{3}+\left(-2a^{5}+15a^{3}+10a^{2}-10a-5\right){x}^{2}+\left(8a^{5}-2a^{4}-57a^{3}-28a^{2}+34a+11\right){x}-8a^{5}+2a^{4}+57a^{3}+28a^{2}-34a-11$ |
| 1.1-a3 |
1.1-a |
$4$ |
$39$ |
6.6.434581.1 |
$6$ |
$[6, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$58.90795$ |
$\textsf{none}$ |
0 |
$\Z/13\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
✓ |
$3, 13$ |
3B, 13B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$71447.18768$ |
0.641303 |
\( 7824861 a^{5} - 5363852 a^{4} - 38334375 a^{3} - 11300754 a^{2} + 16416844 a + 5946574 \) |
\( \bigl[3 a^{5} - 7 a^{4} - 8 a^{3} + 15 a^{2} + a - 4\) , \( a^{5} - 3 a^{4} - 2 a^{3} + 8 a^{2} - 3\) , \( a^{5} - 3 a^{4} - 2 a^{3} + 8 a^{2} - 2\) , \( -3 a^{4} + 4 a^{3} + 12 a^{2} - 3 a - 4\) , \( 10 a^{5} - 28 a^{4} - 19 a^{3} + 67 a^{2} - 8 a - 15\bigr] \) |
${y}^2+\left(3a^{5}-7a^{4}-8a^{3}+15a^{2}+a-4\right){x}{y}+\left(a^{5}-3a^{4}-2a^{3}+8a^{2}-2\right){y}={x}^{3}+\left(a^{5}-3a^{4}-2a^{3}+8a^{2}-3\right){x}^{2}+\left(-3a^{4}+4a^{3}+12a^{2}-3a-4\right){x}+10a^{5}-28a^{4}-19a^{3}+67a^{2}-8a-15$ |
| 1.1-a4 |
1.1-a |
$4$ |
$39$ |
6.6.434581.1 |
$6$ |
$[6, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$58.90795$ |
$\textsf{none}$ |
0 |
$\Z/13\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
✓ |
$3, 13$ |
3B, 13B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$71447.18768$ |
0.641303 |
\( -23419364 a^{5} + 64630767 a^{4} + 44556569 a^{3} - 150905488 a^{2} + 21033374 a + 30775655 \) |
\( \bigl[2 a^{5} - 4 a^{4} - 7 a^{3} + 8 a^{2} + 4 a - 1\) , \( a^{5} - 3 a^{4} - 2 a^{3} + 8 a^{2} - 4\) , \( a^{4} - 2 a^{3} - 3 a^{2} + 3 a + 1\) , \( -3 a^{5} + 4 a^{4} + 15 a^{3} - 6 a^{2} - 17 a - 3\) , \( 3 a^{5} - 4 a^{4} - 15 a^{3} + 5 a^{2} + 16 a + 4\bigr] \) |
${y}^2+\left(2a^{5}-4a^{4}-7a^{3}+8a^{2}+4a-1\right){x}{y}+\left(a^{4}-2a^{3}-3a^{2}+3a+1\right){y}={x}^{3}+\left(a^{5}-3a^{4}-2a^{3}+8a^{2}-4\right){x}^{2}+\left(-3a^{5}+4a^{4}+15a^{3}-6a^{2}-17a-3\right){x}+3a^{5}-4a^{4}-15a^{3}+5a^{2}+16a+4$ |
| 27.1-b2 |
27.1-b |
$2$ |
$13$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{6} \) |
$79.22201$ |
$(a^3+a^2-2a-1)$ |
$1$ |
$\Z/13\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$13$ |
13B.1.1 |
$1$ |
\( 2 \) |
$0.076270034$ |
$226777.4369$ |
1.82314 |
\( -\frac{28672}{3} \) |
\( \bigl[0\) , \( -a^{5} + 5 a^{3} - a^{2} - 5 a + 2\) , \( a^{5} - 5 a^{3} + a^{2} + 5 a - 2\) , \( -a^{5} + 7 a^{3} - a^{2} - 11 a + 2\) , \( a^{5} - 7 a^{3} + a^{2} + 11 a - 2\bigr] \) |
${y}^2+\left(a^{5}-5a^{3}+a^{2}+5a-2\right){y}={x}^{3}+\left(-a^{5}+5a^{3}-a^{2}-5a+2\right){x}^{2}+\left(-a^{5}+7a^{3}-a^{2}-11a+2\right){x}+a^{5}-7a^{3}+a^{2}+11a-2$ |
| 64.1-b1 |
64.1-b |
$2$ |
$13$ |
6.6.820125.1 |
$6$ |
$[6, 0]$ |
64.1 |
\( 2^{6} \) |
\( 2^{12} \) |
$114.44414$ |
$(2)$ |
$2$ |
$\Z/13\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$13$ |
13B.1.1 |
$1$ |
\( 2 \) |
$0.042799640$ |
$234382.7732$ |
4.71924 |
\( -\frac{1339893}{4} \) |
\( \bigl[\frac{18}{19} a^{5} + \frac{7}{19} a^{4} - \frac{154}{19} a^{3} - \frac{134}{19} a^{2} + \frac{93}{19} a + \frac{68}{19}\) , \( \frac{6}{19} a^{5} + \frac{15}{19} a^{4} - \frac{64}{19} a^{3} - \frac{146}{19} a^{2} + \frac{50}{19} a + \frac{67}{19}\) , \( \frac{21}{19} a^{5} + \frac{5}{19} a^{4} - \frac{186}{19} a^{3} - \frac{131}{19} a^{2} + \frac{156}{19} a + \frac{92}{19}\) , \( \frac{73}{19} a^{5} + \frac{59}{19} a^{4} - \frac{614}{19} a^{3} - \frac{782}{19} a^{2} + \frac{89}{19} a + \frac{185}{19}\) , \( -\frac{46}{19} a^{5} - \frac{134}{19} a^{4} + \frac{364}{19} a^{3} + \frac{1246}{19} a^{2} + \frac{554}{19} a - \frac{7}{19}\bigr] \) |
${y}^2+\left(\frac{18}{19}a^{5}+\frac{7}{19}a^{4}-\frac{154}{19}a^{3}-\frac{134}{19}a^{2}+\frac{93}{19}a+\frac{68}{19}\right){x}{y}+\left(\frac{21}{19}a^{5}+\frac{5}{19}a^{4}-\frac{186}{19}a^{3}-\frac{131}{19}a^{2}+\frac{156}{19}a+\frac{92}{19}\right){y}={x}^{3}+\left(\frac{6}{19}a^{5}+\frac{15}{19}a^{4}-\frac{64}{19}a^{3}-\frac{146}{19}a^{2}+\frac{50}{19}a+\frac{67}{19}\right){x}^{2}+\left(\frac{73}{19}a^{5}+\frac{59}{19}a^{4}-\frac{614}{19}a^{3}-\frac{782}{19}a^{2}+\frac{89}{19}a+\frac{185}{19}\right){x}-\frac{46}{19}a^{5}-\frac{134}{19}a^{4}+\frac{364}{19}a^{3}+\frac{1246}{19}a^{2}+\frac{554}{19}a-\frac{7}{19}$ |
| 106.1-c2 |
106.1-c |
$2$ |
$13$ |
6.6.1868969.1 |
$6$ |
$[6, 0]$ |
106.1 |
\( 2 \cdot 53 \) |
\( - 2^{13} \cdot 53 \) |
$180.18353$ |
$(a), (-a^5+a^4+5a^3-3a^2-6a+1)$ |
0 |
$\Z/13\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$13$ |
13B.1.1 |
$1$ |
\( 13 \) |
$1$ |
$73978.80697$ |
4.16258 |
\( -\frac{8560749803}{434176} a^{5} - \frac{31916311079}{217088} a^{4} + \frac{39467641755}{217088} a^{3} + \frac{339746365255}{434176} a^{2} - \frac{58601338489}{217088} a - \frac{307458186015}{434176} \) |
\( \bigl[a^{3} - 2 a - 1\) , \( -a^{4} + a^{3} + 4 a^{2} - a - 2\) , \( a^{5} - 5 a^{3} - a^{2} + 5 a\) , \( 2 a^{5} - a^{4} - 9 a^{3} + 4 a^{2} + 10 a - 4\) , \( -a^{5} + 2 a^{4} + 7 a^{3} - 5 a^{2} - 8 a + 4\bigr] \) |
${y}^2+\left(a^{3}-2a-1\right){x}{y}+\left(a^{5}-5a^{3}-a^{2}+5a\right){y}={x}^{3}+\left(-a^{4}+a^{3}+4a^{2}-a-2\right){x}^{2}+\left(2a^{5}-a^{4}-9a^{3}+4a^{2}+10a-4\right){x}-a^{5}+2a^{4}+7a^{3}-5a^{2}-8a+4$ |
*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.
