| 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 |
| 71.4-a1 |
71.4-a |
$2$ |
$2$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.4 |
\( 71 \) |
\( - 71^{5} \) |
$69.83308$ |
$(6a^5-2a^4-43a^3-17a^2+29a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$2697.010303$ |
1.23075 |
\( \frac{704803031197900}{1804229351} a^{5} - \frac{2010453848133504}{1804229351} a^{4} - \frac{2574645568787336}{1804229351} a^{3} + \frac{8015459823309236}{1804229351} a^{2} - \frac{2156284877910108}{1804229351} a - \frac{1240401241894705}{1804229351} \) |
\( \bigl[-4 a^{5} + a^{4} + 29 a^{3} + 14 a^{2} - 19 a - 6\) , \( 11 a^{5} - 3 a^{4} - 79 a^{3} - 36 a^{2} + 52 a + 17\) , \( -5 a^{5} + a^{4} + 37 a^{3} + 18 a^{2} - 27 a - 8\) , \( -74 a^{5} + 22 a^{4} + 533 a^{3} + 231 a^{2} - 348 a - 101\) , \( -794 a^{5} + 224 a^{4} + 5719 a^{3} + 2523 a^{2} - 3738 a - 1101\bigr] \) |
${y}^2+\left(-4a^{5}+a^{4}+29a^{3}+14a^{2}-19a-6\right){x}{y}+\left(-5a^{5}+a^{4}+37a^{3}+18a^{2}-27a-8\right){y}={x}^{3}+\left(11a^{5}-3a^{4}-79a^{3}-36a^{2}+52a+17\right){x}^{2}+\left(-74a^{5}+22a^{4}+533a^{3}+231a^{2}-348a-101\right){x}-794a^{5}+224a^{4}+5719a^{3}+2523a^{2}-3738a-1101$ |
| 71.4-a2 |
71.4-a |
$2$ |
$2$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.4 |
\( 71 \) |
\( - 71^{10} \) |
$69.83308$ |
$(6a^5-2a^4-43a^3-17a^2+29a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$1348.505151$ |
1.23075 |
\( \frac{21967998164354285461425122}{3255243551009881201} a^{5} - \frac{5391052931075206605671520}{3255243551009881201} a^{4} - \frac{156246015653998240659421204}{3255243551009881201} a^{3} - \frac{69717312288925414565855022}{3255243551009881201} a^{2} + \frac{102032514074835709609288768}{3255243551009881201} a + \frac{30222798799898190818601583}{3255243551009881201} \) |
\( \bigl[a\) , \( -a^{5} + a^{4} + 7 a^{3} - 2 a^{2} - 7 a + 3\) , \( -6 a^{5} + a^{4} + 44 a^{3} + 23 a^{2} - 29 a - 10\) , \( 61 a^{5} - 13 a^{4} - 437 a^{3} - 222 a^{2} + 251 a + 75\) , \( 5699 a^{5} - 1304 a^{4} - 40900 a^{3} - 20142 a^{2} + 24368 a + 7391\bigr] \) |
${y}^2+a{x}{y}+\left(-6a^{5}+a^{4}+44a^{3}+23a^{2}-29a-10\right){y}={x}^{3}+\left(-a^{5}+a^{4}+7a^{3}-2a^{2}-7a+3\right){x}^{2}+\left(61a^{5}-13a^{4}-437a^{3}-222a^{2}+251a+75\right){x}+5699a^{5}-1304a^{4}-40900a^{3}-20142a^{2}+24368a+7391$ |
| 71.4-b1 |
71.4-b |
$2$ |
$2$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.4 |
\( 71 \) |
\( -71 \) |
$69.83308$ |
$(6a^5-2a^4-43a^3-17a^2+29a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.003038782$ |
$267854.1885$ |
2.22863 |
\( -\frac{4285612}{71} a^{5} + \frac{728978}{71} a^{4} + \frac{31362943}{71} a^{3} + \frac{16332660}{71} a^{2} - \frac{20322774}{71} a - \frac{6420718}{71} \) |
\( \bigl[-3 a^{5} + a^{4} + 22 a^{3} + 8 a^{2} - 16 a - 2\) , \( -4 a^{5} + 30 a^{3} + 19 a^{2} - 19 a - 7\) , \( -3 a^{5} + a^{4} + 22 a^{3} + 8 a^{2} - 16 a - 2\) , \( -5 a^{5} + 37 a^{3} + 25 a^{2} - 23 a - 11\) , \( -a^{5} + 7 a^{3} + 6 a^{2} - 4 a - 3\bigr] \) |
${y}^2+\left(-3a^{5}+a^{4}+22a^{3}+8a^{2}-16a-2\right){x}{y}+\left(-3a^{5}+a^{4}+22a^{3}+8a^{2}-16a-2\right){y}={x}^{3}+\left(-4a^{5}+30a^{3}+19a^{2}-19a-7\right){x}^{2}+\left(-5a^{5}+37a^{3}+25a^{2}-23a-11\right){x}-a^{5}+7a^{3}+6a^{2}-4a-3$ |
| 71.4-b2 |
71.4-b |
$2$ |
$2$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.4 |
\( 71 \) |
\( - 71^{2} \) |
$69.83308$ |
$(6a^5-2a^4-43a^3-17a^2+29a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.006077565$ |
$66963.54714$ |
2.22863 |
\( \frac{2634370822463}{5041} a^{5} - \frac{740439838413}{5041} a^{4} - \frac{19177442690303}{5041} a^{3} - \frac{8090290411651}{5041} a^{2} + \frac{13243147162542}{5041} a + \frac{3945985807101}{5041} \) |
\( \bigl[-3 a^{5} + 23 a^{3} + 14 a^{2} - 16 a - 5\) , \( -4 a^{5} + a^{4} + 29 a^{3} + 14 a^{2} - 21 a - 7\) , \( a^{5} - 7 a^{3} - 5 a^{2} + 3 a + 3\) , \( 23 a^{5} - 17 a^{4} - 160 a^{3} + 3 a^{2} + 116 a - 21\) , \( -39 a^{5} + 37 a^{4} + 268 a^{3} - 63 a^{2} - 227 a + 94\bigr] \) |
${y}^2+\left(-3a^{5}+23a^{3}+14a^{2}-16a-5\right){x}{y}+\left(a^{5}-7a^{3}-5a^{2}+3a+3\right){y}={x}^{3}+\left(-4a^{5}+a^{4}+29a^{3}+14a^{2}-21a-7\right){x}^{2}+\left(23a^{5}-17a^{4}-160a^{3}+3a^{2}+116a-21\right){x}-39a^{5}+37a^{4}+268a^{3}-63a^{2}-227a+94$ |
| 71.4-c1 |
71.4-c |
$4$ |
$6$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.4 |
\( 71 \) |
\( - 71^{2} \) |
$69.83308$ |
$(6a^5-2a^4-43a^3-17a^2+29a+5)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$12723.95949$ |
1.29032 |
\( \frac{347779040612919069}{5041} a^{5} - \frac{735086186487273612}{5041} a^{4} - \frac{1615818198289203963}{5041} a^{3} + \frac{2495028141702981708}{5041} a^{2} - \frac{344156570897129140}{5041} a - \frac{312285111948345602}{5041} \) |
\( \bigl[-a^{5} + 8 a^{3} + 4 a^{2} - 7 a - 1\) , \( -6 a^{5} + a^{4} + 44 a^{3} + 23 a^{2} - 29 a - 11\) , \( -a^{5} + 8 a^{3} + 4 a^{2} - 7 a\) , \( 24 a^{5} - 63 a^{4} - 44 a^{3} + 68 a^{2} + 5 a - 6\) , \( 297 a^{5} - 821 a^{4} - 439 a^{3} + 939 a^{2} - 86 a - 98\bigr] \) |
${y}^2+\left(-a^{5}+8a^{3}+4a^{2}-7a-1\right){x}{y}+\left(-a^{5}+8a^{3}+4a^{2}-7a\right){y}={x}^{3}+\left(-6a^{5}+a^{4}+44a^{3}+23a^{2}-29a-11\right){x}^{2}+\left(24a^{5}-63a^{4}-44a^{3}+68a^{2}+5a-6\right){x}+297a^{5}-821a^{4}-439a^{3}+939a^{2}-86a-98$ |
| 71.4-c2 |
71.4-c |
$4$ |
$6$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.4 |
\( 71 \) |
\( - 71^{3} \) |
$69.83308$ |
$(6a^5-2a^4-43a^3-17a^2+29a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$81$ |
\( 1 \) |
$1$ |
$34.90798215$ |
1.29032 |
\( -\frac{10394847966890026874}{357911} a^{5} + \frac{2257127006944677009}{357911} a^{4} + \frac{74885814522798141547}{357911} a^{3} + \frac{37038415740266095494}{357911} a^{2} - \frac{44705744028911194336}{357911} a - \frac{13569453503150492597}{357911} \) |
\( \bigl[-8 a^{5} + 2 a^{4} + 58 a^{3} + 27 a^{2} - 38 a - 11\) , \( -a^{5} + 8 a^{3} + 5 a^{2} - 9 a - 4\) , \( -a^{5} + 8 a^{3} + 5 a^{2} - 7 a - 2\) , \( 24 a^{5} - 21 a^{4} - 145 a^{3} - 27 a^{2} + 88 a + 6\) , \( 11 a^{5} - 92 a^{4} + 101 a^{3} + 251 a^{2} - 84 a - 106\bigr] \) |
${y}^2+\left(-8a^{5}+2a^{4}+58a^{3}+27a^{2}-38a-11\right){x}{y}+\left(-a^{5}+8a^{3}+5a^{2}-7a-2\right){y}={x}^{3}+\left(-a^{5}+8a^{3}+5a^{2}-9a-4\right){x}^{2}+\left(24a^{5}-21a^{4}-145a^{3}-27a^{2}+88a+6\right){x}+11a^{5}-92a^{4}+101a^{3}+251a^{2}-84a-106$ |
| 71.4-c3 |
71.4-c |
$4$ |
$6$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.4 |
\( 71 \) |
\( -71 \) |
$69.83308$ |
$(6a^5-2a^4-43a^3-17a^2+29a+5)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$25447.91898$ |
1.29032 |
\( \frac{178368935}{71} a^{5} - \frac{383017370}{71} a^{4} - \frac{820587590}{71} a^{3} + \frac{1313585339}{71} a^{2} - \frac{193267801}{71} a - \frac{167117569}{71} \) |
\( \bigl[-8 a^{5} + 2 a^{4} + 58 a^{3} + 27 a^{2} - 38 a - 11\) , \( -a^{5} + 8 a^{3} + 5 a^{2} - 9 a - 4\) , \( -a^{5} + 8 a^{3} + 5 a^{2} - 7 a - 2\) , \( -16 a^{5} + 4 a^{4} + 115 a^{3} + 53 a^{2} - 72 a - 19\) , \( 5 a^{5} - a^{4} - 36 a^{3} - 20 a^{2} + 21 a + 8\bigr] \) |
${y}^2+\left(-8a^{5}+2a^{4}+58a^{3}+27a^{2}-38a-11\right){x}{y}+\left(-a^{5}+8a^{3}+5a^{2}-7a-2\right){y}={x}^{3}+\left(-a^{5}+8a^{3}+5a^{2}-9a-4\right){x}^{2}+\left(-16a^{5}+4a^{4}+115a^{3}+53a^{2}-72a-19\right){x}+5a^{5}-a^{4}-36a^{3}-20a^{2}+21a+8$ |
| 71.4-c4 |
71.4-c |
$4$ |
$6$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.4 |
\( 71 \) |
\( - 71^{6} \) |
$69.83308$ |
$(6a^5-2a^4-43a^3-17a^2+29a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$81$ |
\( 2 \) |
$1$ |
$17.45399107$ |
1.29032 |
\( -\frac{122473963803363478904490388871087231}{128100283921} a^{5} + \frac{369584216037450793046353949556548449}{128100283921} a^{4} + \frac{111624163999057123131195127865875451}{128100283921} a^{3} - \frac{470167022871026850042855444550668562}{128100283921} a^{2} + \frac{91317277554701667937618786737072378}{128100283921} a + \frac{60701131070426909908677598532343306}{128100283921} \) |
\( \bigl[-3 a^{5} + 23 a^{3} + 14 a^{2} - 16 a - 6\) , \( -3 a^{5} + 23 a^{3} + 14 a^{2} - 16 a - 5\) , \( a^{5} - 7 a^{3} - 5 a^{2} + 2 a + 3\) , \( -42 a^{5} - 18 a^{4} + 323 a^{3} + 285 a^{2} - 118 a - 210\) , \( 84 a^{5} - 417 a^{4} - 158 a^{3} + 1621 a^{2} + 169 a - 898\bigr] \) |
${y}^2+\left(-3a^{5}+23a^{3}+14a^{2}-16a-6\right){x}{y}+\left(a^{5}-7a^{3}-5a^{2}+2a+3\right){y}={x}^{3}+\left(-3a^{5}+23a^{3}+14a^{2}-16a-5\right){x}^{2}+\left(-42a^{5}-18a^{4}+323a^{3}+285a^{2}-118a-210\right){x}+84a^{5}-417a^{4}-158a^{3}+1621a^{2}+169a-898$ |
*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.
