| 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 |
| 71.1-a1 |
71.1-a |
$4$ |
$6$ |
6.6.434581.1 |
$6$ |
$[6, 0]$ |
71.1 |
\( 71 \) |
\( -71 \) |
$84.03215$ |
$(2a^5-6a^4-4a^3+17a^2-a-6)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$1$ |
$3633.522533$ |
1.37795 |
\( \frac{297477349}{71} a^{5} - \frac{837358584}{71} a^{4} - \frac{502474921}{71} a^{3} + \frac{1866551783}{71} a^{2} - \frac{278037049}{71} a - \frac{384949468}{71} \) |
\( \bigl[a^{5} - 3 a^{4} - 2 a^{3} + 8 a^{2} + a - 2\) , \( 2 a^{5} - 4 a^{4} - 7 a^{3} + 9 a^{2} + 2 a - 3\) , \( 2 a^{5} - 4 a^{4} - 6 a^{3} + 7 a^{2} + 2 a - 1\) , \( 3 a^{5} - 10 a^{4} - 6 a^{3} + 31 a^{2} + 3 a - 14\) , \( 4 a^{5} - 9 a^{4} - 13 a^{3} + 22 a^{2} + 7 a - 10\bigr] \) |
${y}^2+\left(a^{5}-3a^{4}-2a^{3}+8a^{2}+a-2\right){x}{y}+\left(2a^{5}-4a^{4}-6a^{3}+7a^{2}+2a-1\right){y}={x}^{3}+\left(2a^{5}-4a^{4}-7a^{3}+9a^{2}+2a-3\right){x}^{2}+\left(3a^{5}-10a^{4}-6a^{3}+31a^{2}+3a-14\right){x}+4a^{5}-9a^{4}-13a^{3}+22a^{2}+7a-10$ |
| 71.1-a2 |
71.1-a |
$4$ |
$6$ |
6.6.434581.1 |
$6$ |
$[6, 0]$ |
71.1 |
\( 71 \) |
\( - 71^{3} \) |
$84.03215$ |
$(2a^5-6a^4-4a^3+17a^2-a-6)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$1$ |
$3633.522533$ |
1.37795 |
\( -\frac{191935393}{357911} a^{5} - \frac{580390420}{357911} a^{4} + \frac{436997224}{357911} a^{3} + \frac{785354329}{357911} a^{2} - \frac{527031693}{357911} a + \frac{334971338}{357911} \) |
\( \bigl[a^{5} - 2 a^{4} - 4 a^{3} + 5 a^{2} + 4 a - 1\) , \( -a^{5} + 6 a^{3} + 4 a^{2} - 3 a - 2\) , \( 0\) , \( -4 a^{5} + 3 a^{4} + 19 a^{3} + 5 a^{2} - 7 a - 2\) , \( 0\bigr] \) |
${y}^2+\left(a^{5}-2a^{4}-4a^{3}+5a^{2}+4a-1\right){x}{y}={x}^{3}+\left(-a^{5}+6a^{3}+4a^{2}-3a-2\right){x}^{2}+\left(-4a^{5}+3a^{4}+19a^{3}+5a^{2}-7a-2\right){x}$ |
| 71.1-a3 |
71.1-a |
$4$ |
$6$ |
6.6.434581.1 |
$6$ |
$[6, 0]$ |
71.1 |
\( 71 \) |
\( - 71^{6} \) |
$84.03215$ |
$(2a^5-6a^4-4a^3+17a^2-a-6)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$1816.761266$ |
1.37795 |
\( \frac{14623190787592690023}{128100283921} a^{5} + \frac{9262831648934631201}{128100283921} a^{4} - \frac{34215233218956422760}{128100283921} a^{3} - \frac{17202576339060202666}{128100283921} a^{2} + \frac{13171892585884276804}{128100283921} a + \frac{5541195868726527748}{128100283921} \) |
\( \bigl[a^{5} - 2 a^{4} - 4 a^{3} + 5 a^{2} + 3 a - 2\) , \( a^{5} - 3 a^{4} - a^{3} + 6 a^{2} - 3 a - 2\) , \( 2 a^{5} - 4 a^{4} - 6 a^{3} + 7 a^{2} + 2 a - 1\) , \( 10 a^{5} - 30 a^{4} - 10 a^{3} + 51 a^{2} - a - 14\) , \( -15 a^{5} + 55 a^{4} - 31 a^{3} - 39 a^{2} + 17 a + 6\bigr] \) |
${y}^2+\left(a^{5}-2a^{4}-4a^{3}+5a^{2}+3a-2\right){x}{y}+\left(2a^{5}-4a^{4}-6a^{3}+7a^{2}+2a-1\right){y}={x}^{3}+\left(a^{5}-3a^{4}-a^{3}+6a^{2}-3a-2\right){x}^{2}+\left(10a^{5}-30a^{4}-10a^{3}+51a^{2}-a-14\right){x}-15a^{5}+55a^{4}-31a^{3}-39a^{2}+17a+6$ |
| 71.1-a4 |
71.1-a |
$4$ |
$6$ |
6.6.434581.1 |
$6$ |
$[6, 0]$ |
71.1 |
\( 71 \) |
\( - 71^{2} \) |
$84.03215$ |
$(2a^5-6a^4-4a^3+17a^2-a-6)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$1816.761266$ |
1.37795 |
\( -\frac{7067022792014335}{5041} a^{5} + \frac{24612619416466217}{5041} a^{4} - \frac{8158647124759995}{5041} a^{3} - \frac{23426232526371070}{5041} a^{2} + \frac{6414105640727850}{5041} a + \frac{4812827440073893}{5041} \) |
\( \bigl[a^{5} - 2 a^{4} - 3 a^{3} + 3 a^{2} + a + 1\) , \( -2 a^{5} + 5 a^{4} + 5 a^{3} - 11 a^{2} - a + 2\) , \( a^{5} - 2 a^{4} - 3 a^{3} + 3 a^{2} + a\) , \( -97 a^{5} + 157 a^{4} + 417 a^{3} - 259 a^{2} - 387 a - 94\) , \( -599 a^{5} + 877 a^{4} + 2755 a^{3} - 1288 a^{2} - 2719 a - 723\bigr] \) |
${y}^2+\left(a^{5}-2a^{4}-3a^{3}+3a^{2}+a+1\right){x}{y}+\left(a^{5}-2a^{4}-3a^{3}+3a^{2}+a\right){y}={x}^{3}+\left(-2a^{5}+5a^{4}+5a^{3}-11a^{2}-a+2\right){x}^{2}+\left(-97a^{5}+157a^{4}+417a^{3}-259a^{2}-387a-94\right){x}-599a^{5}+877a^{4}+2755a^{3}-1288a^{2}-2719a-723$ |
| 71.1-b1 |
71.1-b |
$2$ |
$3$ |
6.6.434581.1 |
$6$ |
$[6, 0]$ |
71.1 |
\( 71 \) |
\( 71^{3} \) |
$84.03215$ |
$(2a^5-6a^4-4a^3+17a^2-a-6)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$81$ |
\( 1 \) |
$1$ |
$9.122861674$ |
1.12094 |
\( -\frac{25415776248234932727}{357911} a^{5} + \frac{33644549177505288315}{357911} a^{4} + \frac{124407707776619205840}{357911} a^{3} - \frac{42935695592915487066}{357911} a^{2} - \frac{130674592083511357736}{357911} a - \frac{37564287471550457576}{357911} \) |
\( \bigl[3 a^{5} - 6 a^{4} - 10 a^{3} + 12 a^{2} + 4 a - 2\) , \( -a^{5} + 6 a^{3} + 4 a^{2} - 5 a - 3\) , \( a^{5} - 2 a^{4} - 3 a^{3} + 4 a^{2} - 1\) , \( -297 a^{5} + 798 a^{4} + 604 a^{3} - 1840 a^{2} + 208 a + 366\) , \( -3433 a^{5} + 9442 a^{4} + 6587 a^{3} - 22010 a^{2} + 3003 a + 4476\bigr] \) |
${y}^2+\left(3a^{5}-6a^{4}-10a^{3}+12a^{2}+4a-2\right){x}{y}+\left(a^{5}-2a^{4}-3a^{3}+4a^{2}-1\right){y}={x}^{3}+\left(-a^{5}+6a^{3}+4a^{2}-5a-3\right){x}^{2}+\left(-297a^{5}+798a^{4}+604a^{3}-1840a^{2}+208a+366\right){x}-3433a^{5}+9442a^{4}+6587a^{3}-22010a^{2}+3003a+4476$ |
| 71.1-b2 |
71.1-b |
$2$ |
$3$ |
6.6.434581.1 |
$6$ |
$[6, 0]$ |
71.1 |
\( 71 \) |
\( 71 \) |
$84.03215$ |
$(2a^5-6a^4-4a^3+17a^2-a-6)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$6650.566160$ |
1.12094 |
\( \frac{5253651}{71} a^{5} - \frac{13681584}{71} a^{4} - \frac{14298038}{71} a^{3} + \frac{37562313}{71} a^{2} + \frac{4849167}{71} a - \frac{18566036}{71} \) |
\( \bigl[2 a^{5} - 4 a^{4} - 7 a^{3} + 8 a^{2} + 4 a - 1\) , \( -2 a^{4} + 3 a^{3} + 7 a^{2} - 3 a - 4\) , \( 3 a^{5} - 6 a^{4} - 10 a^{3} + 12 a^{2} + 5 a - 3\) , \( -3 a^{5} + 6 a^{4} + 9 a^{3} - 9 a^{2} - 3 a\) , \( -a^{5} + 2 a^{4} + 3 a^{3} - 4 a^{2} + a - 1\bigr] \) |
${y}^2+\left(2a^{5}-4a^{4}-7a^{3}+8a^{2}+4a-1\right){x}{y}+\left(3a^{5}-6a^{4}-10a^{3}+12a^{2}+5a-3\right){y}={x}^{3}+\left(-2a^{4}+3a^{3}+7a^{2}-3a-4\right){x}^{2}+\left(-3a^{5}+6a^{4}+9a^{3}-9a^{2}-3a\right){x}-a^{5}+2a^{4}+3a^{3}-4a^{2}+a-1$ |
*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.
