| 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 |
| 25.2-a1 |
25.2-a |
$1$ |
$1$ |
6.6.1241125.1 |
$6$ |
$[6, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{3} \) |
$130.17912$ |
$(-2a^5+a^4+13a^3-2a^2-19a-5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$0.008964453$ |
$31253.00479$ |
3.01779 |
\( -2708 a^{5} + 733 a^{4} + 12894 a^{3} - 3948 a^{2} - 9886 a - 1842 \) |
\( \bigl[-a^{5} + 7 a^{3} + 2 a^{2} - 10 a - 6\) , \( 2 a^{5} - a^{4} - 13 a^{3} + 2 a^{2} + 20 a + 5\) , \( -5 a^{5} + 2 a^{4} + 34 a^{3} - 2 a^{2} - 53 a - 19\) , \( -6 a^{5} + 2 a^{4} + 40 a^{3} - 60 a - 21\) , \( -5 a^{5} + a^{4} + 35 a^{3} + 3 a^{2} - 55 a - 24\bigr] \) |
${y}^2+\left(-a^{5}+7a^{3}+2a^{2}-10a-6\right){x}{y}+\left(-5a^{5}+2a^{4}+34a^{3}-2a^{2}-53a-19\right){y}={x}^{3}+\left(2a^{5}-a^{4}-13a^{3}+2a^{2}+20a+5\right){x}^{2}+\left(-6a^{5}+2a^{4}+40a^{3}-60a-21\right){x}-5a^{5}+a^{4}+35a^{3}+3a^{2}-55a-24$ |
| 25.2-b1 |
25.2-b |
$4$ |
$15$ |
6.6.1241125.1 |
$6$ |
$[6, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{21} \) |
$130.17912$ |
$(-2a^5+a^4+13a^3-2a^2-19a-5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 5$ |
3B, 5B.1.4[2] |
$25$ |
\( 2 \) |
$1$ |
$40.42142472$ |
1.81415 |
\( -\frac{584205938671037576}{390625} a^{5} + \frac{127830640643771423}{390625} a^{4} + \frac{4061470830963324523}{390625} a^{3} + \frac{279717696988069228}{390625} a^{2} - \frac{1297494123233689326}{78125} a - \frac{2669912142068228202}{390625} \) |
\( \bigl[-a^{5} + a^{4} + 6 a^{3} - 3 a^{2} - 8 a - 3\) , \( -6 a^{5} + 2 a^{4} + 41 a^{3} - a^{2} - 65 a - 23\) , \( -3 a^{5} + a^{4} + 20 a^{3} - 29 a - 12\) , \( -45 a^{5} + 22 a^{4} + 303 a^{3} - 59 a^{2} - 463 a - 84\) , \( -5610 a^{5} + 2584 a^{4} + 38079 a^{3} - 6320 a^{2} - 58796 a - 12189\bigr] \) |
${y}^2+\left(-a^{5}+a^{4}+6a^{3}-3a^{2}-8a-3\right){x}{y}+\left(-3a^{5}+a^{4}+20a^{3}-29a-12\right){y}={x}^{3}+\left(-6a^{5}+2a^{4}+41a^{3}-a^{2}-65a-23\right){x}^{2}+\left(-45a^{5}+22a^{4}+303a^{3}-59a^{2}-463a-84\right){x}-5610a^{5}+2584a^{4}+38079a^{3}-6320a^{2}-58796a-12189$ |
| 25.2-b2 |
25.2-b |
$4$ |
$15$ |
6.6.1241125.1 |
$6$ |
$[6, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{7} \) |
$130.17912$ |
$(-2a^5+a^4+13a^3-2a^2-19a-5)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 5$ |
3B, 5B.1.1[2] |
$1$ |
\( 2 \) |
$1$ |
$25263.39045$ |
1.81415 |
\( \frac{1332544}{5} a^{5} + \frac{2260678}{5} a^{4} - \frac{5421687}{5} a^{3} - \frac{11771837}{5} a^{2} - 1128428 a - \frac{732232}{5} \) |
\( \bigl[-a^{5} + 7 a^{3} + 2 a^{2} - 11 a - 6\) , \( -3 a^{5} + a^{4} + 20 a^{3} - 30 a - 12\) , \( -5 a^{5} + 2 a^{4} + 34 a^{3} - 2 a^{2} - 52 a - 19\) , \( -5 a^{5} + a^{4} + 36 a^{3} + 3 a^{2} - 62 a - 25\) , \( -6 a^{5} + a^{4} + 41 a^{3} + 4 a^{2} - 64 a - 27\bigr] \) |
${y}^2+\left(-a^{5}+7a^{3}+2a^{2}-11a-6\right){x}{y}+\left(-5a^{5}+2a^{4}+34a^{3}-2a^{2}-52a-19\right){y}={x}^{3}+\left(-3a^{5}+a^{4}+20a^{3}-30a-12\right){x}^{2}+\left(-5a^{5}+a^{4}+36a^{3}+3a^{2}-62a-25\right){x}-6a^{5}+a^{4}+41a^{3}+4a^{2}-64a-27$ |
| 25.2-b3 |
25.2-b |
$4$ |
$15$ |
6.6.1241125.1 |
$6$ |
$[6, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{11} \) |
$130.17912$ |
$(-2a^5+a^4+13a^3-2a^2-19a-5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 5$ |
3B, 5B.1.4[2] |
$25$ |
\( 2 \) |
$1$ |
$40.42142472$ |
1.81415 |
\( -\frac{13681459876}{125} a^{5} - \frac{36187576902}{125} a^{4} + \frac{25331233773}{125} a^{3} + \frac{99876956003}{125} a^{2} + \frac{10695502109}{25} a + \frac{7259236048}{125} \) |
\( \bigl[-2 a^{5} + a^{4} + 14 a^{3} - a^{2} - 23 a - 10\) , \( 4 a^{5} - a^{4} - 28 a^{3} - a^{2} + 45 a + 16\) , \( -5 a^{5} + 2 a^{4} + 34 a^{3} - 2 a^{2} - 53 a - 18\) , \( 4 a^{5} - a^{4} - 48 a^{3} - 8 a^{2} + 139 a + 64\) , \( 5 a^{5} + 11 a^{4} - 60 a^{3} - 66 a^{2} + 162 a + 80\bigr] \) |
${y}^2+\left(-2a^{5}+a^{4}+14a^{3}-a^{2}-23a-10\right){x}{y}+\left(-5a^{5}+2a^{4}+34a^{3}-2a^{2}-53a-18\right){y}={x}^{3}+\left(4a^{5}-a^{4}-28a^{3}-a^{2}+45a+16\right){x}^{2}+\left(4a^{5}-a^{4}-48a^{3}-8a^{2}+139a+64\right){x}+5a^{5}+11a^{4}-60a^{3}-66a^{2}+162a+80$ |
| 25.2-b4 |
25.2-b |
$4$ |
$15$ |
6.6.1241125.1 |
$6$ |
$[6, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{9} \) |
$130.17912$ |
$(-2a^5+a^4+13a^3-2a^2-19a-5)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 5$ |
3B, 5B.1.1[2] |
$1$ |
\( 2 \) |
$1$ |
$25263.39045$ |
1.81415 |
\( \frac{3288568389824}{25} a^{5} - \frac{1515811489972}{25} a^{4} - \frac{22321293042632}{25} a^{3} + \frac{3711500587103}{25} a^{2} + \frac{6892701907756}{5} a + \frac{7134572445643}{25} \) |
\( \bigl[-3 a^{5} + 2 a^{4} + 20 a^{3} - 5 a^{2} - 30 a - 10\) , \( -6 a^{5} + 2 a^{4} + 41 a^{3} - a^{2} - 65 a - 21\) , \( -3 a^{5} + a^{4} + 20 a^{3} - 29 a - 12\) , \( -2 a^{5} - 3 a^{4} + 18 a^{3} + 16 a^{2} - 36 a - 16\) , \( -2 a^{5} - 2 a^{4} + 17 a^{3} + 16 a^{2} - 40 a - 20\bigr] \) |
${y}^2+\left(-3a^{5}+2a^{4}+20a^{3}-5a^{2}-30a-10\right){x}{y}+\left(-3a^{5}+a^{4}+20a^{3}-29a-12\right){y}={x}^{3}+\left(-6a^{5}+2a^{4}+41a^{3}-a^{2}-65a-21\right){x}^{2}+\left(-2a^{5}-3a^{4}+18a^{3}+16a^{2}-36a-16\right){x}-2a^{5}-2a^{4}+17a^{3}+16a^{2}-40a-20$ |
| 25.2-c1 |
25.2-c |
$1$ |
$1$ |
6.6.1241125.1 |
$6$ |
$[6, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{19} \) |
$130.17912$ |
$(-2a^5+a^4+13a^3-2a^2-19a-5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$808.4057645$ |
1.45128 |
\( \frac{1376405109}{78125} a^{5} - \frac{1325860482}{78125} a^{4} - \frac{7597745332}{78125} a^{3} + \frac{2698464798}{78125} a^{2} + \frac{2247897599}{15625} a + \frac{2321410818}{78125} \) |
\( \bigl[-2 a^{5} + a^{4} + 13 a^{3} - 2 a^{2} - 18 a - 6\) , \( -a^{2} + a + 2\) , \( -a^{5} + 7 a^{3} + 2 a^{2} - 11 a - 5\) , \( 26 a^{5} - 12 a^{4} - 176 a^{3} + 29 a^{2} + 270 a + 59\) , \( -51 a^{5} + 24 a^{4} + 346 a^{3} - 61 a^{2} - 534 a - 106\bigr] \) |
${y}^2+\left(-2a^{5}+a^{4}+13a^{3}-2a^{2}-18a-6\right){x}{y}+\left(-a^{5}+7a^{3}+2a^{2}-11a-5\right){y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(26a^{5}-12a^{4}-176a^{3}+29a^{2}+270a+59\right){x}-51a^{5}+24a^{4}+346a^{3}-61a^{2}-534a-106$ |
| 25.2-d1 |
25.2-d |
$1$ |
$1$ |
6.6.1241125.1 |
$6$ |
$[6, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{9} \) |
$130.17912$ |
$(-2a^5+a^4+13a^3-2a^2-19a-5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$0.020207192$ |
$12955.98893$ |
2.82001 |
\( -2708 a^{5} + 733 a^{4} + 12894 a^{3} - 3948 a^{2} - 9886 a - 1842 \) |
\( \bigl[a^{2} + a - 3\) , \( a^{4} - a^{3} - 5 a^{2} + 3 a + 3\) , \( -5 a^{5} + 2 a^{4} + 34 a^{3} - 2 a^{2} - 52 a - 19\) , \( -8 a^{5} + 3 a^{4} + 53 a^{3} - 5 a^{2} - 79 a - 22\) , \( -8 a^{5} + 3 a^{4} + 54 a^{3} - 4 a^{2} - 83 a - 27\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(-5a^{5}+2a^{4}+34a^{3}-2a^{2}-52a-19\right){y}={x}^{3}+\left(a^{4}-a^{3}-5a^{2}+3a+3\right){x}^{2}+\left(-8a^{5}+3a^{4}+53a^{3}-5a^{2}-79a-22\right){x}-8a^{5}+3a^{4}+54a^{3}-4a^{2}-83a-27$ |
*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.
