| 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 |
| 25.1-a1 |
25.1-a |
$4$ |
$4$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{4} \) |
$23.55381$ |
$(a^4-2a^3-2a^2+5a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$1769.528882$ |
1.15781477 |
\( \frac{708112}{25} a^{4} - \frac{1125079}{25} a^{3} - \frac{2581192}{25} a^{2} + \frac{496376}{5} a + \frac{1711623}{25} \) |
\( \bigl[2 a^{4} - 3 a^{3} - 6 a^{2} + 7 a + 1\) , \( -2 a^{4} + 3 a^{3} + 8 a^{2} - 7 a - 5\) , \( a^{4} - 2 a^{3} - 2 a^{2} + 4 a - 1\) , \( 2 a^{4} - 2 a^{3} - 4 a^{2} + 4 a\) , \( 2 a^{4} - a^{3} - 5 a^{2} + 2 a\bigr] \) |
${y}^2+\left(2a^{4}-3a^{3}-6a^{2}+7a+1\right){x}{y}+\left(a^{4}-2a^{3}-2a^{2}+4a-1\right){y}={x}^{3}+\left(-2a^{4}+3a^{3}+8a^{2}-7a-5\right){x}^{2}+\left(2a^{4}-2a^{3}-4a^{2}+4a\right){x}+2a^{4}-a^{3}-5a^{2}+2a$ |
| 25.1-a2 |
25.1-a |
$4$ |
$4$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{8} \) |
$23.55381$ |
$(a^4-2a^3-2a^2+5a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$884.7644411$ |
1.15781477 |
\( \frac{1549868991089}{625} a^{4} - \frac{2463604541892}{625} a^{3} - \frac{5660845609864}{625} a^{2} + \frac{5425940335044}{625} a + \frac{3777390596864}{625} \) |
\( \bigl[a^{4} - 2 a^{3} - 3 a^{2} + 5 a + 2\) , \( a^{4} - a^{3} - 5 a^{2} + 4 a + 4\) , \( a^{4} - 2 a^{3} - 3 a^{2} + 6 a + 2\) , \( 20 a^{4} - 18 a^{3} - 76 a^{2} + 10 a + 20\) , \( 69 a^{4} - 48 a^{3} - 268 a^{2} - 6 a + 54\bigr] \) |
${y}^2+\left(a^{4}-2a^{3}-3a^{2}+5a+2\right){x}{y}+\left(a^{4}-2a^{3}-3a^{2}+6a+2\right){y}={x}^{3}+\left(a^{4}-a^{3}-5a^{2}+4a+4\right){x}^{2}+\left(20a^{4}-18a^{3}-76a^{2}+10a+20\right){x}+69a^{4}-48a^{3}-268a^{2}-6a+54$ |
| 25.1-a3 |
25.1-a |
$4$ |
$4$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
25.1 |
\( 5^{2} \) |
\( - 5^{16} \) |
$23.55381$ |
$(a^4-2a^3-2a^2+5a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1$ |
$27.64888878$ |
1.15781477 |
\( -\frac{136402392772743238}{390625} a^{4} + \frac{341339187700279242}{390625} a^{3} + \frac{236740655090238193}{390625} a^{2} - \frac{799831738588792786}{390625} a + \frac{268640638721913956}{390625} \) |
\( \bigl[a^{4} - 2 a^{3} - 3 a^{2} + 6 a + 1\) , \( -a^{4} + 2 a^{3} + 4 a^{2} - 5 a - 4\) , \( a^{4} - a^{3} - 4 a^{2} + 3 a + 2\) , \( -5 a^{4} - 14 a^{3} - 11 a^{2} + 55 a - 20\) , \( -63 a^{4} - 115 a^{3} + 166 a^{2} + 193 a - 100\bigr] \) |
${y}^2+\left(a^{4}-2a^{3}-3a^{2}+6a+1\right){x}{y}+\left(a^{4}-a^{3}-4a^{2}+3a+2\right){y}={x}^{3}+\left(-a^{4}+2a^{3}+4a^{2}-5a-4\right){x}^{2}+\left(-5a^{4}-14a^{3}-11a^{2}+55a-20\right){x}-63a^{4}-115a^{3}+166a^{2}+193a-100$ |
| 25.1-a4 |
25.1-a |
$4$ |
$4$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{4} \) |
$23.55381$ |
$(a^4-2a^3-2a^2+5a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$442.3822205$ |
1.15781477 |
\( \frac{488509962822731359638}{25} a^{4} - \frac{776547463172625188346}{25} a^{3} - \frac{1784205847864588683433}{25} a^{2} + \frac{342071130235394451034}{5} a + \frac{1190397818850374480652}{25} \) |
\( \bigl[a^{4} - a^{3} - 3 a^{2} + 2 a + 1\) , \( -a^{4} + 2 a^{3} + 3 a^{2} - 4 a - 2\) , \( a^{2} - 1\) , \( -53 a^{4} + 93 a^{3} + 155 a^{2} - 135 a - 149\) , \( -157 a^{4} + 112 a^{3} + 991 a^{2} - 666 a - 644\bigr] \) |
${y}^2+\left(a^{4}-a^{3}-3a^{2}+2a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(-a^{4}+2a^{3}+3a^{2}-4a-2\right){x}^{2}+\left(-53a^{4}+93a^{3}+155a^{2}-135a-149\right){x}-157a^{4}+112a^{3}+991a^{2}-666a-644$ |
| 25.1-b1 |
25.1-b |
$1$ |
$1$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{2} \) |
$23.55381$ |
$(a^4-2a^3-2a^2+5a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$424.5187979$ |
2.22212427 |
\( -\frac{1476597}{5} a^{4} + \frac{213601}{5} a^{3} + \frac{3235067}{5} a^{2} - \frac{577692}{5} a - \frac{972922}{5} \) |
\( \bigl[a^{4} - a^{3} - 4 a^{2} + 3 a + 2\) , \( -a^{4} + 2 a^{3} + 4 a^{2} - 5 a - 3\) , \( a^{4} - a^{3} - 3 a^{2} + a + 1\) , \( -28 a^{4} + 44 a^{3} + 105 a^{2} - 98 a - 72\) , \( 137 a^{4} - 218 a^{3} - 500 a^{2} + 480 a + 332\bigr] \) |
${y}^2+\left(a^{4}-a^{3}-4a^{2}+3a+2\right){x}{y}+\left(a^{4}-a^{3}-3a^{2}+a+1\right){y}={x}^{3}+\left(-a^{4}+2a^{3}+4a^{2}-5a-3\right){x}^{2}+\left(-28a^{4}+44a^{3}+105a^{2}-98a-72\right){x}+137a^{4}-218a^{3}-500a^{2}+480a+332$ |
| 25.1-c1 |
25.1-c |
$1$ |
$1$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{2} \) |
$23.55381$ |
$(a^4-2a^3-2a^2+5a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.001191585$ |
$36862.41352$ |
1.14960935 |
\( -\frac{1476597}{5} a^{4} + \frac{213601}{5} a^{3} + \frac{3235067}{5} a^{2} - \frac{577692}{5} a - \frac{972922}{5} \) |
\( \bigl[a^{4} - 2 a^{3} - 3 a^{2} + 5 a + 1\) , \( -a^{4} + 2 a^{3} + 3 a^{2} - 6 a - 1\) , \( 2 a^{4} - 3 a^{3} - 6 a^{2} + 6 a + 1\) , \( -2 a^{4} + 4 a^{3} + 8 a^{2} - 11 a - 8\) , \( 2 a^{4} - 6 a^{3} - 6 a^{2} + 17 a + 7\bigr] \) |
${y}^2+\left(a^{4}-2a^{3}-3a^{2}+5a+1\right){x}{y}+\left(2a^{4}-3a^{3}-6a^{2}+6a+1\right){y}={x}^{3}+\left(-a^{4}+2a^{3}+3a^{2}-6a-1\right){x}^{2}+\left(-2a^{4}+4a^{3}+8a^{2}-11a-8\right){x}+2a^{4}-6a^{3}-6a^{2}+17a+7$ |
| 25.1-d1 |
25.1-d |
$4$ |
$4$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{4} \) |
$23.55381$ |
$(a^4-2a^3-2a^2+5a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.200199043$ |
$2693.592912$ |
1.76419015 |
\( \frac{708112}{25} a^{4} - \frac{1125079}{25} a^{3} - \frac{2581192}{25} a^{2} + \frac{496376}{5} a + \frac{1711623}{25} \) |
\( \bigl[a^{2} - 2\) , \( a^{4} - 2 a^{3} - 4 a^{2} + 6 a + 3\) , \( a^{2} - 1\) , \( 3 a^{4} - 6 a^{3} - 10 a^{2} + 13 a + 6\) , \( -20 a^{4} + 12 a^{3} + 77 a^{2} + 9 a - 11\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{4}-2a^{3}-4a^{2}+6a+3\right){x}^{2}+\left(3a^{4}-6a^{3}-10a^{2}+13a+6\right){x}-20a^{4}+12a^{3}+77a^{2}+9a-11$ |
| 25.1-d2 |
25.1-d |
$4$ |
$4$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{8} \) |
$23.55381$ |
$(a^4-2a^3-2a^2+5a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$0.400398087$ |
$1346.796456$ |
1.76419015 |
\( \frac{1549868991089}{625} a^{4} - \frac{2463604541892}{625} a^{3} - \frac{5660845609864}{625} a^{2} + \frac{5425940335044}{625} a + \frac{3777390596864}{625} \) |
\( \bigl[2 a^{4} - 3 a^{3} - 7 a^{2} + 7 a + 3\) , \( -a^{2} + 1\) , \( a^{4} - 2 a^{3} - 2 a^{2} + 5 a - 1\) , \( -8 a^{4} - 21 a^{3} + 15 a^{2} + 40 a - 7\) , \( -51 a^{4} - 10 a^{3} + 111 a^{2} - a - 9\bigr] \) |
${y}^2+\left(2a^{4}-3a^{3}-7a^{2}+7a+3\right){x}{y}+\left(a^{4}-2a^{3}-2a^{2}+5a-1\right){y}={x}^{3}+\left(-a^{2}+1\right){x}^{2}+\left(-8a^{4}-21a^{3}+15a^{2}+40a-7\right){x}-51a^{4}-10a^{3}+111a^{2}-a-9$ |
| 25.1-d3 |
25.1-d |
$4$ |
$4$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
25.1 |
\( 5^{2} \) |
\( - 5^{16} \) |
$23.55381$ |
$(a^4-2a^3-2a^2+5a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.200199043$ |
$673.3982280$ |
1.76419015 |
\( -\frac{136402392772743238}{390625} a^{4} + \frac{341339187700279242}{390625} a^{3} + \frac{236740655090238193}{390625} a^{2} - \frac{799831738588792786}{390625} a + \frac{268640638721913956}{390625} \) |
\( \bigl[a^{4} - a^{3} - 3 a^{2} + 2 a\) , \( -a^{4} + 2 a^{3} + 2 a^{2} - 3 a + 1\) , \( a^{2} - 2\) , \( 89 a^{4} - 17 a^{3} - 439 a^{2} - 38 a + 86\) , \( -819 a^{4} + 586 a^{3} + 3095 a^{2} + 134 a - 596\bigr] \) |
${y}^2+\left(a^{4}-a^{3}-3a^{2}+2a\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{4}+2a^{3}+2a^{2}-3a+1\right){x}^{2}+\left(89a^{4}-17a^{3}-439a^{2}-38a+86\right){x}-819a^{4}+586a^{3}+3095a^{2}+134a-596$ |
| 25.1-d4 |
25.1-d |
$4$ |
$4$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{4} \) |
$23.55381$ |
$(a^4-2a^3-2a^2+5a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$0.800796175$ |
$42.08738925$ |
1.76419015 |
\( \frac{488509962822731359638}{25} a^{4} - \frac{776547463172625188346}{25} a^{3} - \frac{1784205847864588683433}{25} a^{2} + \frac{342071130235394451034}{5} a + \frac{1190397818850374480652}{25} \) |
\( \bigl[a^{2} - 2\) , \( a^{4} - 2 a^{3} - 4 a^{2} + 6 a + 3\) , \( a^{2} - 1\) , \( 143 a^{4} - 101 a^{3} - 555 a^{2} + 8 a + 86\) , \( 747 a^{4} - 457 a^{3} - 2953 a^{2} - 247 a + 559\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{4}-2a^{3}-4a^{2}+6a+3\right){x}^{2}+\left(143a^{4}-101a^{3}-555a^{2}+8a+86\right){x}+747a^{4}-457a^{3}-2953a^{2}-247a+559$ |
*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.
