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 |
71.3-a1 |
71.3-a |
$1$ |
$1$ |
4.4.6125.1 |
$4$ |
$[4, 0]$ |
71.3 |
\( 71 \) |
\( 71 \) |
$11.91513$ |
$(5a^3+7a^2-28a-24)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.880646448$ |
$82.10556773$ |
3.695567517 |
\( \frac{14499836867}{71} a^{3} - \frac{55487831122}{71} a^{2} + \frac{26243447485}{71} a + \frac{56492779821}{71} \) |
\( \bigl[2 a^{3} + 3 a^{2} - 11 a - 12\) , \( -a^{3} - a^{2} + 6 a + 4\) , \( a^{3} + a^{2} - 5 a - 3\) , \( -a^{2} - 2 a + 1\) , \( -2 a^{3} - 4 a^{2} + 8 a + 7\bigr] \) |
${y}^2+\left(2a^{3}+3a^{2}-11a-12\right){x}{y}+\left(a^{3}+a^{2}-5a-3\right){y}={x}^{3}+\left(-a^{3}-a^{2}+6a+4\right){x}^{2}+\left(-a^{2}-2a+1\right){x}-2a^{3}-4a^{2}+8a+7$ |
71.3-b1 |
71.3-b |
$1$ |
$1$ |
4.4.6125.1 |
$4$ |
$[4, 0]$ |
71.3 |
\( 71 \) |
\( 71^{3} \) |
$11.91513$ |
$(5a^3+7a^2-28a-24)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3 \) |
$0.278940957$ |
$74.56956939$ |
3.189349568 |
\( \frac{171505521527}{357911} a^{3} + \frac{230646737948}{357911} a^{2} - \frac{982547272228}{357911} a - \frac{796389209025}{357911} \) |
\( \bigl[1\) , \( -a^{2} - a + 5\) , \( a^{3} + 2 a^{2} - 5 a - 7\) , \( a^{3} - 8 a^{2} + 5 a + 17\) , \( 3 a^{3} - 24 a^{2} + 24 a + 30\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}+2a^{2}-5a-7\right){y}={x}^{3}+\left(-a^{2}-a+5\right){x}^{2}+\left(a^{3}-8a^{2}+5a+17\right){x}+3a^{3}-24a^{2}+24a+30$ |
71.3-c1 |
71.3-c |
$1$ |
$1$ |
4.4.6125.1 |
$4$ |
$[4, 0]$ |
71.3 |
\( 71 \) |
\( 71^{2} \) |
$11.91513$ |
$(5a^3+7a^2-28a-24)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1.350817753$ |
$24.25015266$ |
3.348483607 |
\( \frac{13630521344}{5041} a^{3} + \frac{59020271616}{5041} a^{2} - \frac{66721566720}{5041} a - \frac{352246767616}{5041} \) |
\( \bigl[0\) , \( a^{2} - 5\) , \( 1\) , \( a^{3} - 9 a\) , \( 610 a^{3} + 832 a^{2} - 3525 a - 2843\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a^{2}-5\right){x}^{2}+\left(a^{3}-9a\right){x}+610a^{3}+832a^{2}-3525a-2843$ |
71.3-d1 |
71.3-d |
$1$ |
$1$ |
4.4.6125.1 |
$4$ |
$[4, 0]$ |
71.3 |
\( 71 \) |
\( 71 \) |
$11.91513$ |
$(5a^3+7a^2-28a-24)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.077631787$ |
$821.9823011$ |
3.261437020 |
\( \frac{5520752072949}{71} a^{3} + \frac{7531137764446}{71} a^{2} - \frac{31882024190872}{71} a - \frac{25687141194430}{71} \) |
\( \bigl[2 a^{3} + 3 a^{2} - 12 a - 12\) , \( a^{2} + a - 4\) , \( a^{3} + 2 a^{2} - 6 a - 7\) , \( 10 a^{3} + 12 a^{2} - 62 a - 46\) , \( 35 a^{3} + 41 a^{2} - 221 a - 165\bigr] \) |
${y}^2+\left(2a^{3}+3a^{2}-12a-12\right){x}{y}+\left(a^{3}+2a^{2}-6a-7\right){y}={x}^{3}+\left(a^{2}+a-4\right){x}^{2}+\left(10a^{3}+12a^{2}-62a-46\right){x}+35a^{3}+41a^{2}-221a-165$ |
71.3-e1 |
71.3-e |
$1$ |
$1$ |
4.4.6125.1 |
$4$ |
$[4, 0]$ |
71.3 |
\( 71 \) |
\( 71^{3} \) |
$11.91513$ |
$(5a^3+7a^2-28a-24)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3 \) |
$0.012087042$ |
$1375.479221$ |
2.549190457 |
\( \frac{171505521527}{357911} a^{3} + \frac{230646737948}{357911} a^{2} - \frac{982547272228}{357911} a - \frac{796389209025}{357911} \) |
\( \bigl[2 a^{3} + 3 a^{2} - 12 a - 11\) , \( -1\) , \( a^{3} + a^{2} - 5 a - 4\) , \( 12 a^{3} + 15 a^{2} - 77 a - 64\) , \( 24 a^{3} + 28 a^{2} - 152 a - 116\bigr] \) |
${y}^2+\left(2a^{3}+3a^{2}-12a-11\right){x}{y}+\left(a^{3}+a^{2}-5a-4\right){y}={x}^{3}-{x}^{2}+\left(12a^{3}+15a^{2}-77a-64\right){x}+24a^{3}+28a^{2}-152a-116$ |
71.3-f1 |
71.3-f |
$1$ |
$1$ |
4.4.6125.1 |
$4$ |
$[4, 0]$ |
71.3 |
\( 71 \) |
\( 71 \) |
$11.91513$ |
$(5a^3+7a^2-28a-24)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.316585081$ |
$168.4566808$ |
2.725747549 |
\( \frac{5520752072949}{71} a^{3} + \frac{7531137764446}{71} a^{2} - \frac{31882024190872}{71} a - \frac{25687141194430}{71} \) |
\( \bigl[2 a^{3} + 3 a^{2} - 11 a - 11\) , \( 2 a^{3} + 3 a^{2} - 13 a - 13\) , \( a\) , \( 2 a^{3} + 4 a^{2} - 11 a - 8\) , \( 5 a^{3} + 7 a^{2} - 28 a - 33\bigr] \) |
${y}^2+\left(2a^{3}+3a^{2}-11a-11\right){x}{y}+a{y}={x}^{3}+\left(2a^{3}+3a^{2}-13a-13\right){x}^{2}+\left(2a^{3}+4a^{2}-11a-8\right){x}+5a^{3}+7a^{2}-28a-33$ |
71.3-g1 |
71.3-g |
$1$ |
$1$ |
4.4.6125.1 |
$4$ |
$[4, 0]$ |
71.3 |
\( 71 \) |
\( 71 \) |
$11.91513$ |
$(5a^3+7a^2-28a-24)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.021483562$ |
$2332.543676$ |
2.561197211 |
\( \frac{14499836867}{71} a^{3} - \frac{55487831122}{71} a^{2} + \frac{26243447485}{71} a + \frac{56492779821}{71} \) |
\( \bigl[a\) , \( a^{2} - a - 3\) , \( 2 a^{3} + 3 a^{2} - 11 a - 11\) , \( 4 a^{2} - 5 a - 17\) , \( -18 a^{2} - 16 a + 129\bigr] \) |
${y}^2+a{x}{y}+\left(2a^{3}+3a^{2}-11a-11\right){y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(4a^{2}-5a-17\right){x}-18a^{2}-16a+129$ |
71.3-h1 |
71.3-h |
$1$ |
$1$ |
4.4.6125.1 |
$4$ |
$[4, 0]$ |
71.3 |
\( 71 \) |
\( 71^{2} \) |
$11.91513$ |
$(5a^3+7a^2-28a-24)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.021909019$ |
$1115.169704$ |
2.497473303 |
\( \frac{13630521344}{5041} a^{3} + \frac{59020271616}{5041} a^{2} - \frac{66721566720}{5041} a - \frac{352246767616}{5041} \) |
\( \bigl[0\) , \( -a^{3} - a^{2} + 7 a + 5\) , \( a\) , \( 3 a^{2} + 2 a - 19\) , \( 2 a^{3} - 3 a^{2} - 15 a + 25\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-a^{3}-a^{2}+7a+5\right){x}^{2}+\left(3a^{2}+2a-19\right){x}+2a^{3}-3a^{2}-15a+25$ |
*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.