| 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 |
| 19.2-a1 |
19.2-a |
$4$ |
$6$ |
6.6.1397493.1 |
$6$ |
$[6, 0]$ |
19.2 |
\( 19 \) |
\( 19 \) |
$135.01318$ |
$(-a^2+a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 1 \) |
$0.927758489$ |
$24058.42713$ |
3.14685 |
\( \frac{30438798074416398357}{19} a^{5} - \frac{73523888941416578820}{19} a^{4} - \frac{134293589945377162173}{19} a^{3} + \frac{225888842823227670348}{19} a^{2} + \frac{223356050294960948895}{19} a - \frac{52073635140874193409}{19} \) |
\( \bigl[a^{2} - 1\) , \( -a^{3} + 2 a^{2} + 2 a - 3\) , \( a^{5} - 2 a^{4} - 6 a^{3} + 9 a^{2} + 9 a - 5\) , \( 61 a^{5} - 52 a^{4} - 294 a^{3} - 54 a^{2} + 119 a - 22\) , \( -385 a^{5} + 338 a^{4} + 1905 a^{3} + 207 a^{2} - 929 a + 163\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{5}-2a^{4}-6a^{3}+9a^{2}+9a-5\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+2a-3\right){x}^{2}+\left(61a^{5}-52a^{4}-294a^{3}-54a^{2}+119a-22\right){x}-385a^{5}+338a^{4}+1905a^{3}+207a^{2}-929a+163$ |
| 19.2-a2 |
19.2-a |
$4$ |
$6$ |
6.6.1397493.1 |
$6$ |
$[6, 0]$ |
19.2 |
\( 19 \) |
\( 19^{3} \) |
$135.01318$ |
$(-a^2+a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 3 \) |
$2.783275467$ |
$297.0176190$ |
3.14685 |
\( \frac{23356303984655729540145}{6859} a^{5} - \frac{7452712481411092047777}{6859} a^{4} - \frac{90048979498873307140854}{6859} a^{3} - \frac{7850364710050201601979}{6859} a^{2} + \frac{49022773693222173695892}{6859} a - \frac{8712073558822488019641}{6859} \) |
\( \bigl[a^{5} - 2 a^{4} - 6 a^{3} + 8 a^{2} + 10 a - 4\) , \( a^{5} - 3 a^{4} - 2 a^{3} + 8 a^{2} + 2 a - 4\) , \( a^{5} - 3 a^{4} - 2 a^{3} + 7 a^{2} + 3 a\) , \( -47 a^{5} + 97 a^{4} + 278 a^{3} - 375 a^{2} - 479 a + 89\) , \( 342 a^{5} - 795 a^{4} - 1606 a^{3} + 2521 a^{2} + 2619 a - 626\bigr] \) |
${y}^2+\left(a^{5}-2a^{4}-6a^{3}+8a^{2}+10a-4\right){x}{y}+\left(a^{5}-3a^{4}-2a^{3}+7a^{2}+3a\right){y}={x}^{3}+\left(a^{5}-3a^{4}-2a^{3}+8a^{2}+2a-4\right){x}^{2}+\left(-47a^{5}+97a^{4}+278a^{3}-375a^{2}-479a+89\right){x}+342a^{5}-795a^{4}-1606a^{3}+2521a^{2}+2619a-626$ |
| 19.2-a3 |
19.2-a |
$4$ |
$6$ |
6.6.1397493.1 |
$6$ |
$[6, 0]$ |
19.2 |
\( 19 \) |
\( 19^{6} \) |
$135.01318$ |
$(-a^2+a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$1.391637733$ |
$297.0176190$ |
3.14685 |
\( -\frac{12210045787883583}{47045881} a^{5} + \frac{3896591398207569}{47045881} a^{4} + \frac{47074450079680857}{47045881} a^{3} + \frac{4102787318035545}{47045881} a^{2} - \frac{25626747801780036}{47045881} a + \frac{4554281634804891}{47045881} \) |
\( \bigl[a^{5} - 2 a^{4} - 5 a^{3} + 6 a^{2} + 8 a - 2\) , \( -a^{5} + 4 a^{4} - 11 a^{2} + 2 a + 3\) , \( a^{3} - a^{2} - 3 a\) , \( 5 a^{5} - 8 a^{4} - 20 a^{3} + 10 a^{2} + 12 a\) , \( 11 a^{5} - 10 a^{4} - 56 a^{3} - 2 a^{2} + 33 a - 6\bigr] \) |
${y}^2+\left(a^{5}-2a^{4}-5a^{3}+6a^{2}+8a-2\right){x}{y}+\left(a^{3}-a^{2}-3a\right){y}={x}^{3}+\left(-a^{5}+4a^{4}-11a^{2}+2a+3\right){x}^{2}+\left(5a^{5}-8a^{4}-20a^{3}+10a^{2}+12a\right){x}+11a^{5}-10a^{4}-56a^{3}-2a^{2}+33a-6$ |
| 19.2-a4 |
19.2-a |
$4$ |
$6$ |
6.6.1397493.1 |
$6$ |
$[6, 0]$ |
19.2 |
\( 19 \) |
\( 19^{2} \) |
$135.01318$ |
$(-a^2+a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$0.463879244$ |
$24058.42713$ |
3.14685 |
\( -\frac{227915510049}{361} a^{5} + \frac{550521041508}{361} a^{4} + \frac{1005549589422}{361} a^{3} - \frac{1691380479852}{361} a^{2} - \frac{1672422076449}{361} a + \frac{389911279446}{361} \) |
\( \bigl[a^{2} - 1\) , \( -a^{3} + 2 a^{2} + 2 a - 3\) , \( a^{5} - 2 a^{4} - 6 a^{3} + 9 a^{2} + 9 a - 5\) , \( -14 a^{5} + 13 a^{4} + 71 a^{3} + a^{2} - 41 a + 8\) , \( -70 a^{5} + 64 a^{4} + 349 a^{3} + 20 a^{2} - 187 a + 31\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{5}-2a^{4}-6a^{3}+9a^{2}+9a-5\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+2a-3\right){x}^{2}+\left(-14a^{5}+13a^{4}+71a^{3}+a^{2}-41a+8\right){x}-70a^{5}+64a^{4}+349a^{3}+20a^{2}-187a+31$ |
| 19.2-b1 |
19.2-b |
$4$ |
$6$ |
6.6.1397493.1 |
$6$ |
$[6, 0]$ |
19.2 |
\( 19 \) |
\( 19 \) |
$135.01318$ |
$(-a^2+a+1)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
|
\( 1 \) |
$1$ |
$1573.351246$ |
2.82341 |
\( \frac{30438798074416398357}{19} a^{5} - \frac{73523888941416578820}{19} a^{4} - \frac{134293589945377162173}{19} a^{3} + \frac{225888842823227670348}{19} a^{2} + \frac{223356050294960948895}{19} a - \frac{52073635140874193409}{19} \) |
\( \bigl[a^{4} - 3 a^{3} - a^{2} + 6 a - 1\) , \( a^{4} - 4 a^{3} + 9 a - 1\) , \( 0\) , \( -20 a^{5} + 76 a^{4} - 8 a^{3} - 164 a^{2} + 68 a - 8\) , \( 3 a^{5} - 44 a^{4} + 119 a^{3} + 36 a^{2} - 277 a + 52\bigr] \) |
${y}^2+\left(a^{4}-3a^{3}-a^{2}+6a-1\right){x}{y}={x}^{3}+\left(a^{4}-4a^{3}+9a-1\right){x}^{2}+\left(-20a^{5}+76a^{4}-8a^{3}-164a^{2}+68a-8\right){x}+3a^{5}-44a^{4}+119a^{3}+36a^{2}-277a+52$ |
| 19.2-b2 |
19.2-b |
$4$ |
$6$ |
6.6.1397493.1 |
$6$ |
$[6, 0]$ |
19.2 |
\( 19 \) |
\( 19^{3} \) |
$135.01318$ |
$(-a^2+a+1)$ |
$0 \le r \le 1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
|
\( 3 \) |
$1$ |
$14160.16121$ |
2.82341 |
\( \frac{23356303984655729540145}{6859} a^{5} - \frac{7452712481411092047777}{6859} a^{4} - \frac{90048979498873307140854}{6859} a^{3} - \frac{7850364710050201601979}{6859} a^{2} + \frac{49022773693222173695892}{6859} a - \frac{8712073558822488019641}{6859} \) |
\( \bigl[a^{5} - 2 a^{4} - 6 a^{3} + 8 a^{2} + 11 a - 4\) , \( -a^{4} + 4 a^{3} - 2 a^{2} - 7 a + 5\) , \( a^{4} - 2 a^{3} - 3 a^{2} + 5 a + 1\) , \( 2 a^{5} - 104 a^{4} + 135 a^{3} + 202 a^{2} - 111 a - 20\) , \( 368 a^{5} - 56 a^{4} - 1473 a^{3} - 288 a^{2} + 785 a - 89\bigr] \) |
${y}^2+\left(a^{5}-2a^{4}-6a^{3}+8a^{2}+11a-4\right){x}{y}+\left(a^{4}-2a^{3}-3a^{2}+5a+1\right){y}={x}^{3}+\left(-a^{4}+4a^{3}-2a^{2}-7a+5\right){x}^{2}+\left(2a^{5}-104a^{4}+135a^{3}+202a^{2}-111a-20\right){x}+368a^{5}-56a^{4}-1473a^{3}-288a^{2}+785a-89$ |
| 19.2-b3 |
19.2-b |
$4$ |
$6$ |
6.6.1397493.1 |
$6$ |
$[6, 0]$ |
19.2 |
\( 19 \) |
\( 19^{6} \) |
$135.01318$ |
$(-a^2+a+1)$ |
$0 \le r \le 1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
|
\( 2 \cdot 3 \) |
$1$ |
$14160.16121$ |
2.82341 |
\( -\frac{12210045787883583}{47045881} a^{5} + \frac{3896591398207569}{47045881} a^{4} + \frac{47074450079680857}{47045881} a^{3} + \frac{4102787318035545}{47045881} a^{2} - \frac{25626747801780036}{47045881} a + \frac{4554281634804891}{47045881} \) |
\( \bigl[a\) , \( a^{5} - 3 a^{4} - 4 a^{3} + 11 a^{2} + 7 a - 4\) , \( a^{5} - 2 a^{4} - 6 a^{3} + 8 a^{2} + 10 a - 4\) , \( -105 a^{5} + 429 a^{4} - 160 a^{3} - 856 a^{2} + 635 a - 90\) , \( 1119 a^{5} - 4610 a^{4} + 1797 a^{3} + 9200 a^{2} - 6931 a + 1005\bigr] \) |
${y}^2+a{x}{y}+\left(a^{5}-2a^{4}-6a^{3}+8a^{2}+10a-4\right){y}={x}^{3}+\left(a^{5}-3a^{4}-4a^{3}+11a^{2}+7a-4\right){x}^{2}+\left(-105a^{5}+429a^{4}-160a^{3}-856a^{2}+635a-90\right){x}+1119a^{5}-4610a^{4}+1797a^{3}+9200a^{2}-6931a+1005$ |
| 19.2-b4 |
19.2-b |
$4$ |
$6$ |
6.6.1397493.1 |
$6$ |
$[6, 0]$ |
19.2 |
\( 19 \) |
\( 19^{2} \) |
$135.01318$ |
$(-a^2+a+1)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
|
\( 2 \) |
$1$ |
$1573.351246$ |
2.82341 |
\( -\frac{227915510049}{361} a^{5} + \frac{550521041508}{361} a^{4} + \frac{1005549589422}{361} a^{3} - \frac{1691380479852}{361} a^{2} - \frac{1672422076449}{361} a + \frac{389911279446}{361} \) |
\( \bigl[a^{4} - 3 a^{3} - a^{2} + 6 a - 1\) , \( a^{4} - 4 a^{3} + 9 a - 1\) , \( 0\) , \( 5 a^{5} - 19 a^{4} + 2 a^{3} + 41 a^{2} - 17 a + 2\) , \( 0\bigr] \) |
${y}^2+\left(a^{4}-3a^{3}-a^{2}+6a-1\right){x}{y}={x}^{3}+\left(a^{4}-4a^{3}+9a-1\right){x}^{2}+\left(5a^{5}-19a^{4}+2a^{3}+41a^{2}-17a+2\right){x}$ |
*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.
