| 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 |
| 71.6-a1 |
71.6-a |
$2$ |
$2$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.6 |
\( 71 \) |
\( - 71^{10} \) |
$69.83308$ |
$(-a^4+a^3+6a^2-a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$1348.505151$ |
1.23075 |
\( -\frac{168897361086877089738292772}{3255243551009881201} a^{5} + \frac{36590815305412598961751628}{3255243551009881201} a^{4} + \frac{1215010113934663006249461026}{3255243551009881201} a^{3} + \frac{608534409345028127920951294}{3255243551009881201} a^{2} - \frac{728672484398001360842800916}{3255243551009881201} a - \frac{221646481326554190660944727}{3255243551009881201} \) |
\( \bigl[-2 a^{5} + a^{4} + 14 a^{3} + 4 a^{2} - 10 a - 2\) , \( -3 a^{5} + 2 a^{4} + 20 a^{3} + 3 a^{2} - 13 a - 1\) , \( a^{5} - a^{4} - 6 a^{3} + a^{2} + 3 a\) , \( -a^{5} + 20 a^{4} - 20 a^{3} - 104 a^{2} + 46 a + 16\) , \( -403 a^{5} + 171 a^{4} + 2816 a^{3} + 961 a^{2} - 1694 a - 488\bigr] \) |
${y}^2+\left(-2a^{5}+a^{4}+14a^{3}+4a^{2}-10a-2\right){x}{y}+\left(a^{5}-a^{4}-6a^{3}+a^{2}+3a\right){y}={x}^{3}+\left(-3a^{5}+2a^{4}+20a^{3}+3a^{2}-13a-1\right){x}^{2}+\left(-a^{5}+20a^{4}-20a^{3}-104a^{2}+46a+16\right){x}-403a^{5}+171a^{4}+2816a^{3}+961a^{2}-1694a-488$ |
| 71.6-a2 |
71.6-a |
$2$ |
$2$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.6 |
\( 71 \) |
\( - 71^{5} \) |
$69.83308$ |
$(-a^4+a^3+6a^2-a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$2697.010303$ |
1.23075 |
\( \frac{98022664188220}{1804229351} a^{5} + \frac{450615417702264}{1804229351} a^{4} - \frac{1805568365714444}{1804229351} a^{3} - \frac{1538702394143424}{1804229351} a^{2} + \frac{1421852364024048}{1804229351} a + \frac{470449120328339}{1804229351} \) |
\( \bigl[-4 a^{5} + a^{4} + 29 a^{3} + 13 a^{2} - 19 a - 5\) , \( -a^{2} + a + 3\) , \( -6 a^{5} + a^{4} + 44 a^{3} + 23 a^{2} - 29 a - 10\) , \( -42 a^{5} + 38 a^{4} + 266 a^{3} - 15 a^{2} - 116 a - 22\) , \( -177 a^{5} + 241 a^{4} + 1008 a^{3} - 518 a^{2} - 238 a - 6\bigr] \) |
${y}^2+\left(-4a^{5}+a^{4}+29a^{3}+13a^{2}-19a-5\right){x}{y}+\left(-6a^{5}+a^{4}+44a^{3}+23a^{2}-29a-10\right){y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(-42a^{5}+38a^{4}+266a^{3}-15a^{2}-116a-22\right){x}-177a^{5}+241a^{4}+1008a^{3}-518a^{2}-238a-6$ |
| 71.6-b1 |
71.6-b |
$2$ |
$2$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.6 |
\( 71 \) |
\( -71 \) |
$69.83308$ |
$(-a^4+a^3+6a^2-a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.003038782$ |
$267854.1885$ |
2.22863 |
\( \frac{1584785}{71} a^{5} - \frac{365527}{71} a^{4} - \frac{11832625}{71} a^{3} - \frac{4916339}{71} a^{2} + \frac{8890196}{71} a + \frac{1422813}{71} \) |
\( \bigl[-3 a^{5} + a^{4} + 22 a^{3} + 8 a^{2} - 17 a - 3\) , \( a^{5} - a^{4} - 6 a^{3} + a^{2} + a - 2\) , \( -6 a^{5} + a^{4} + 44 a^{3} + 23 a^{2} - 29 a - 10\) , \( -14 a^{5} + 3 a^{4} + 102 a^{3} + 49 a^{2} - 67 a - 19\) , \( -4 a^{5} + a^{4} + 29 a^{3} + 14 a^{2} - 20 a - 6\bigr] \) |
${y}^2+\left(-3a^{5}+a^{4}+22a^{3}+8a^{2}-17a-3\right){x}{y}+\left(-6a^{5}+a^{4}+44a^{3}+23a^{2}-29a-10\right){y}={x}^{3}+\left(a^{5}-a^{4}-6a^{3}+a^{2}+a-2\right){x}^{2}+\left(-14a^{5}+3a^{4}+102a^{3}+49a^{2}-67a-19\right){x}-4a^{5}+a^{4}+29a^{3}+14a^{2}-20a-6$ |
| 71.6-b2 |
71.6-b |
$2$ |
$2$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.6 |
\( 71 \) |
\( - 71^{2} \) |
$69.83308$ |
$(-a^4+a^3+6a^2-a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.006077565$ |
$66963.54714$ |
2.22863 |
\( \frac{1777124588319}{5041} a^{5} - \frac{622636090605}{5041} a^{4} - \frac{12371540609582}{5041} a^{3} - \frac{5367013179437}{5041} a^{2} + \frac{6675565561120}{5041} a + \frac{3442516804683}{5041} \) |
\( \bigl[-3 a^{5} + a^{4} + 22 a^{3} + 8 a^{2} - 17 a - 3\) , \( a^{5} - a^{4} - 6 a^{3} + a^{2} + a - 2\) , \( -6 a^{5} + a^{4} + 44 a^{3} + 23 a^{2} - 29 a - 10\) , \( -19 a^{5} + 3 a^{4} + 137 a^{3} + 74 a^{2} - 87 a - 34\) , \( a^{5} + 4 a^{4} - 7 a^{3} - 23 a^{2} + 15\bigr] \) |
${y}^2+\left(-3a^{5}+a^{4}+22a^{3}+8a^{2}-17a-3\right){x}{y}+\left(-6a^{5}+a^{4}+44a^{3}+23a^{2}-29a-10\right){y}={x}^{3}+\left(a^{5}-a^{4}-6a^{3}+a^{2}+a-2\right){x}^{2}+\left(-19a^{5}+3a^{4}+137a^{3}+74a^{2}-87a-34\right){x}+a^{5}+4a^{4}-7a^{3}-23a^{2}+15$ |
| 71.6-c1 |
71.6-c |
$4$ |
$6$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.6 |
\( 71 \) |
\( -71 \) |
$69.83308$ |
$(-a^4+a^3+6a^2-a-4)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$25447.91898$ |
1.29032 |
\( -\frac{28972963}{71} a^{5} + \frac{96235706}{71} a^{4} + \frac{5646273}{71} a^{3} - \frac{132507744}{71} a^{2} + \frac{40217498}{71} a + \frac{19267976}{71} \) |
\( \bigl[a + 1\) , \( 5 a^{5} - a^{4} - 36 a^{3} - 18 a^{2} + 21 a + 8\) , \( -5 a^{5} + a^{4} + 37 a^{3} + 18 a^{2} - 26 a - 7\) , \( 6 a^{5} - 2 a^{4} - 43 a^{3} - 15 a^{2} + 29 a + 6\) , \( -4 a^{5} + a^{4} + 29 a^{3} + 16 a^{2} - 17 a - 9\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(-5a^{5}+a^{4}+37a^{3}+18a^{2}-26a-7\right){y}={x}^{3}+\left(5a^{5}-a^{4}-36a^{3}-18a^{2}+21a+8\right){x}^{2}+\left(6a^{5}-2a^{4}-43a^{3}-15a^{2}+29a+6\right){x}-4a^{5}+a^{4}+29a^{3}+16a^{2}-17a-9$ |
| 71.6-c2 |
71.6-c |
$4$ |
$6$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.6 |
\( 71 \) |
\( - 71^{2} \) |
$69.83308$ |
$(-a^4+a^3+6a^2-a-4)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$12723.95949$ |
1.29032 |
\( -\frac{61875465519471921}{5041} a^{5} + \frac{186718135324916475}{5041} a^{4} + \frac{56396294925773264}{5041} a^{3} - \frac{237534589193567280}{5041} a^{2} + \frac{46132023113659095}{5041} a + \frac{30668266783208944}{5041} \) |
\( \bigl[-4 a^{5} + a^{4} + 29 a^{3} + 13 a^{2} - 19 a - 5\) , \( 7 a^{5} - 3 a^{4} - 49 a^{3} - 16 a^{2} + 32 a + 5\) , \( -4 a^{5} + a^{4} + 29 a^{3} + 13 a^{2} - 19 a - 5\) , \( 259 a^{5} - 67 a^{4} - 1864 a^{3} - 861 a^{2} + 1180 a + 354\) , \( -8418 a^{5} + 2341 a^{4} + 60616 a^{3} + 26921 a^{2} - 39489 a - 11664\bigr] \) |
${y}^2+\left(-4a^{5}+a^{4}+29a^{3}+13a^{2}-19a-5\right){x}{y}+\left(-4a^{5}+a^{4}+29a^{3}+13a^{2}-19a-5\right){y}={x}^{3}+\left(7a^{5}-3a^{4}-49a^{3}-16a^{2}+32a+5\right){x}^{2}+\left(259a^{5}-67a^{4}-1864a^{3}-861a^{2}+1180a+354\right){x}-8418a^{5}+2341a^{4}+60616a^{3}+26921a^{2}-39489a-11664$ |
| 71.6-c3 |
71.6-c |
$4$ |
$6$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.6 |
\( 71 \) |
\( - 71^{3} \) |
$69.83308$ |
$(-a^4+a^3+6a^2-a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$81$ |
\( 1 \) |
$1$ |
$34.90798215$ |
1.29032 |
\( \frac{11766293016690075047}{357911} a^{5} - \frac{3846624773200897835}{357911} a^{4} - \frac{84429025191782602931}{357911} a^{3} - \frac{33585882902292945592}{357911} a^{2} + \frac{56063574090909894272}{357911} a + \frac{13549412469350476076}{357911} \) |
\( \bigl[-a^{5} + 8 a^{3} + 5 a^{2} - 7 a - 3\) , \( 4 a^{5} - a^{4} - 29 a^{3} - 13 a^{2} + 19 a + 4\) , \( 3 a^{5} - a^{4} - 21 a^{3} - 9 a^{2} + 12 a + 4\) , \( 63 a^{5} - 19 a^{4} - 441 a^{3} - 212 a^{2} + 256 a + 75\) , \( 137 a^{5} - 41 a^{4} - 974 a^{3} - 478 a^{2} + 562 a + 170\bigr] \) |
${y}^2+\left(-a^{5}+8a^{3}+5a^{2}-7a-3\right){x}{y}+\left(3a^{5}-a^{4}-21a^{3}-9a^{2}+12a+4\right){y}={x}^{3}+\left(4a^{5}-a^{4}-29a^{3}-13a^{2}+19a+4\right){x}^{2}+\left(63a^{5}-19a^{4}-441a^{3}-212a^{2}+256a+75\right){x}+137a^{5}-41a^{4}-974a^{3}-478a^{2}+562a+170$ |
| 71.6-c4 |
71.6-c |
$4$ |
$6$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.6 |
\( 71 \) |
\( - 71^{6} \) |
$69.83308$ |
$(-a^4+a^3+6a^2-a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$81$ |
\( 2 \) |
$1$ |
$17.45399107$ |
1.29032 |
\( -\frac{1497407221113091542123705318886825605}{128100283921} a^{5} + \frac{1909270575865547358106962503949723645}{128100283921} a^{4} + \frac{9956703765127135479902609867280817062}{128100283921} a^{3} - \frac{5733415777361561907282977317465673583}{128100283921} a^{2} - \frac{8904868800301824918302580003940486543}{128100283921} a + \frac{5444107517623868650336317646324773214}{128100283921} \) |
\( \bigl[-a^{5} + 8 a^{3} + 5 a^{2} - 7 a - 3\) , \( 4 a^{5} - a^{4} - 29 a^{3} - 13 a^{2} + 19 a + 4\) , \( 3 a^{5} - a^{4} - 21 a^{3} - 9 a^{2} + 12 a + 4\) , \( 83 a^{5} - 79 a^{4} - 516 a^{3} - 117 a^{2} + 306 a + 30\) , \( -75 a^{5} + 338 a^{4} - 891 a^{3} - 920 a^{2} + 793 a + 141\bigr] \) |
${y}^2+\left(-a^{5}+8a^{3}+5a^{2}-7a-3\right){x}{y}+\left(3a^{5}-a^{4}-21a^{3}-9a^{2}+12a+4\right){y}={x}^{3}+\left(4a^{5}-a^{4}-29a^{3}-13a^{2}+19a+4\right){x}^{2}+\left(83a^{5}-79a^{4}-516a^{3}-117a^{2}+306a+30\right){x}-75a^{5}+338a^{4}-891a^{3}-920a^{2}+793a+141$ |
*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.
