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 |
841.8-a1 |
841.8-a |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
841.8 |
\( 29^{2} \) |
\( 29^{6} \) |
$6.95528$ |
$(2a^3+a^2-7a)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-15$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2 \) |
$1$ |
$103.6460095$ |
1.545063486 |
\( 85995 a^{3} - 257985 a - 52515 \) |
\( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( a^{2} + a - 3\) , \( a^{3} + a^{2} - 2 a - 1\) , \( 732862 a^{3} + 606139 a^{2} - 1823977 a - 401107\) , \( 617507703 a^{3} + 510735017 a^{2} - 1536871501 a - 337973166\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-1\right){x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(a^{2}+a-3\right){x}^{2}+\left(732862a^{3}+606139a^{2}-1823977a-401107\right){x}+617507703a^{3}+510735017a^{2}-1536871501a-337973166$ |
841.8-a2 |
841.8-a |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
841.8 |
\( 29^{2} \) |
\( 29^{6} \) |
$6.95528$ |
$(2a^3+a^2-7a)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-60$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2 \) |
$1$ |
$103.6460095$ |
1.545063486 |
\( -16554983445 a^{3} + 49664950335 a + 10231546590 \) |
\( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( a^{2} + a - 3\) , \( a^{3} + a^{2} - 2 a - 1\) , \( -4944643 a^{3} - 4089726 a^{2} + 12306253 a + 2706273\) , \( 6260338923 a^{3} + 5177869806 a^{2} - 15580916689 a - 3426396906\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-1\right){x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(a^{2}+a-3\right){x}^{2}+\left(-4944643a^{3}-4089726a^{2}+12306253a+2706273\right){x}+6260338923a^{3}+5177869806a^{2}-15580916689a-3426396906$ |
841.8-a3 |
841.8-a |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
841.8 |
\( 29^{2} \) |
\( 29^{6} \) |
$6.95528$ |
$(2a^3+a^2-7a)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-60$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2 \) |
$1$ |
$103.6460095$ |
1.545063486 |
\( -16554983445 a^{3} + 49664950335 a + 10231546590 \) |
\( \bigl[a^{3} - 3 a\) , \( -a^{2} + a + 2\) , \( a^{2} + a - 2\) , \( 46 a^{3} + 3 a^{2} - 179 a - 37\) , \( 315 a^{3} + 91 a^{2} - 1189 a - 251\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(46a^{3}+3a^{2}-179a-37\right){x}+315a^{3}+91a^{2}-1189a-251$ |
841.8-a4 |
841.8-a |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
841.8 |
\( 29^{2} \) |
\( 29^{6} \) |
$6.95528$ |
$(2a^3+a^2-7a)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-15$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2 \) |
$1$ |
$103.6460095$ |
1.545063486 |
\( 85995 a^{3} - 257985 a - 52515 \) |
\( \bigl[a^{3} - 3 a\) , \( -a^{2} + a + 2\) , \( a^{2} + a - 2\) , \( 6 a^{3} + 3 a^{2} - 19 a - 2\) , \( 3 a^{3} - a^{2} - 16 a - 4\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(6a^{3}+3a^{2}-19a-2\right){x}+3a^{3}-a^{2}-16a-4$ |
841.8-a5 |
841.8-a |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
841.8 |
\( 29^{2} \) |
\( 29^{6} \) |
$6.95528$ |
$(2a^3+a^2-7a)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-15$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2 \) |
$1$ |
$103.6460095$ |
1.545063486 |
\( -85995 a^{3} + 257985 a - 138510 \) |
\( \bigl[a\) , \( a^{2} - a - 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -378019 a^{3} - 312648 a^{2} + 940838 a + 206900\) , \( -174410939 a^{3} - 144253725 a^{2} + 434079076 a + 95458261\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(-378019a^{3}-312648a^{2}+940838a+206900\right){x}-174410939a^{3}-144253725a^{2}+434079076a+95458261$ |
841.8-a6 |
841.8-a |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
841.8 |
\( 29^{2} \) |
\( 29^{6} \) |
$6.95528$ |
$(2a^3+a^2-7a)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-60$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2 \) |
$1$ |
$103.6460095$ |
1.545063486 |
\( 16554983445 a^{3} - 49664950335 a + 26786530035 \) |
\( \bigl[a\) , \( a^{2} - a - 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -6055524 a^{3} - 5008513 a^{2} + 15071068 a + 3314280\) , \( -11131060964 a^{3} - 9206399587 a^{2} + 27703314743 a + 6092231498\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(-6055524a^{3}-5008513a^{2}+15071068a+3314280\right){x}-11131060964a^{3}-9206399587a^{2}+27703314743a+6092231498$ |
841.8-a7 |
841.8-a |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
841.8 |
\( 29^{2} \) |
\( 29^{6} \) |
$6.95528$ |
$(2a^3+a^2-7a)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-15$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2 \) |
$1$ |
$103.6460095$ |
1.545063486 |
\( -85995 a^{3} + 257985 a - 138510 \) |
\( \bigl[1\) , \( -a^{3} + a^{2} + 3 a - 3\) , \( a^{2} + a - 2\) , \( -70 a^{3} - 57 a^{2} + 174 a + 37\) , \( 432 a^{3} + 358 a^{2} - 1075 a - 238\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-3\right){x}^{2}+\left(-70a^{3}-57a^{2}+174a+37\right){x}+432a^{3}+358a^{2}-1075a-238$ |
841.8-a8 |
841.8-a |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
841.8 |
\( 29^{2} \) |
\( 29^{6} \) |
$6.95528$ |
$(2a^3+a^2-7a)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-60$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2 \) |
$1$ |
$103.6460095$ |
1.545063486 |
\( 16554983445 a^{3} - 49664950335 a + 26786530035 \) |
\( \bigl[1\) , \( -a^{3} + a^{2} + 3 a - 3\) , \( a^{2} + a - 2\) , \( -1125 a^{3} - 897 a^{2} + 2819 a + 527\) , \( 27556 a^{3} + 22887 a^{2} - 68523 a - 15332\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-3\right){x}^{2}+\left(-1125a^{3}-897a^{2}+2819a+527\right){x}+27556a^{3}+22887a^{2}-68523a-15332$ |
841.8-b1 |
841.8-b |
$2$ |
$3$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
841.8 |
\( 29^{2} \) |
\( 29^{10} \) |
$6.95528$ |
$(2a^3+a^2-7a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$47.68821090$ |
1.421787750 |
\( 0 \) |
\( \bigl[0\) , \( -a^{3} + a^{2} + 3 a - 2\) , \( a^{2} - 2\) , \( -a + 2\) , \( 133196725 a^{3} + 110165536 a^{2} - 331504541 a - 72901094\bigr] \) |
${y}^2+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-2\right){x}^{2}+\left(-a+2\right){x}+133196725a^{3}+110165536a^{2}-331504541a-72901094$ |
841.8-b2 |
841.8-b |
$2$ |
$3$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
841.8 |
\( 29^{2} \) |
\( 29^{10} \) |
$6.95528$ |
$(2a^3+a^2-7a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$47.68821090$ |
1.421787750 |
\( 0 \) |
\( \bigl[0\) , \( a^{2} + a - 1\) , \( a^{3} + a^{2} - 2 a - 2\) , \( a^{3} + a^{2} - 2 a\) , \( -32 a^{3} - 20 a^{2} + 94 a + 20\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-2a-2\right){y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(a^{3}+a^{2}-2a\right){x}-32a^{3}-20a^{2}+94a+20$ |
*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.