| 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 |
$2$ |
$2$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.3 |
\( 71 \) |
\( - 71^{5} \) |
$69.83308$ |
$(-a^5+a^4+7a^3-2a^2-7a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$2697.010303$ |
1.23075 |
\( \frac{28618507660868}{1804229351} a^{5} + \frac{72913497392328}{1804229351} a^{4} - \frac{19705446541660}{1804229351} a^{3} - \frac{57325413156020}{1804229351} a^{2} + \frac{151711549020}{25411681} a + \frac{4220073576031}{1804229351} \) |
\( \bigl[a\) , \( -7 a^{5} + 3 a^{4} + 49 a^{3} + 16 a^{2} - 32 a - 5\) , \( -a^{5} + 8 a^{3} + 5 a^{2} - 7 a - 3\) , \( -a^{5} - a^{4} + 9 a^{3} + 8 a^{2} - 4 a - 2\) , \( 2 a^{5} - 10 a^{4} + 2 a^{3} + 30 a^{2} - 16\bigr] \) |
${y}^2+a{x}{y}+\left(-a^{5}+8a^{3}+5a^{2}-7a-3\right){y}={x}^{3}+\left(-7a^{5}+3a^{4}+49a^{3}+16a^{2}-32a-5\right){x}^{2}+\left(-a^{5}-a^{4}+9a^{3}+8a^{2}-4a-2\right){x}+2a^{5}-10a^{4}+2a^{3}+30a^{2}-16$ |
| 71.3-a2 |
71.3-a |
$2$ |
$2$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.3 |
\( 71 \) |
\( - 71^{10} \) |
$69.83308$ |
$(-a^5+a^4+7a^3-2a^2-7a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$1348.505151$ |
1.23075 |
\( \frac{7245137537592799459668364}{3255243551009881201} a^{5} - \frac{17175864433733831866928686}{3255243551009881201} a^{4} - \frac{29002126063391338539879632}{3255243551009881201} a^{3} + \frac{59053137968044220795988828}{3255243551009881201} a^{2} - \frac{333346305063766203121436}{45848500718449031} a + \frac{536537380242287614298139}{3255243551009881201} \) |
\( \bigl[3 a^{5} - a^{4} - 21 a^{3} - 9 a^{2} + 13 a + 4\) , \( -2 a^{5} + a^{4} + 14 a^{3} + 4 a^{2} - 11 a - 1\) , \( -3 a^{5} + 23 a^{3} + 14 a^{2} - 17 a - 6\) , \( -7 a^{5} - 2 a^{4} + 55 a^{3} + 44 a^{2} - 32 a - 25\) , \( 88 a^{5} - 195 a^{4} - 254 a^{3} + 186 a^{2} + 98 a - 25\bigr] \) |
${y}^2+\left(3a^{5}-a^{4}-21a^{3}-9a^{2}+13a+4\right){x}{y}+\left(-3a^{5}+23a^{3}+14a^{2}-17a-6\right){y}={x}^{3}+\left(-2a^{5}+a^{4}+14a^{3}+4a^{2}-11a-1\right){x}^{2}+\left(-7a^{5}-2a^{4}+55a^{3}+44a^{2}-32a-25\right){x}+88a^{5}-195a^{4}-254a^{3}+186a^{2}+98a-25$ |
| 71.3-b1 |
71.3-b |
$2$ |
$2$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.3 |
\( 71 \) |
\( - 71^{2} \) |
$69.83308$ |
$(-a^5+a^4+7a^3-2a^2-7a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.006077565$ |
$66963.54714$ |
2.22863 |
\( -\frac{2806479973455}{5041} a^{5} + \frac{1209517831721}{5041} a^{4} + \frac{19764644332781}{5041} a^{3} + \frac{6136368206763}{5041} a^{2} - \frac{178823365318}{71} a - \frac{1119026433909}{5041} \) |
\( \bigl[-6 a^{5} + a^{4} + 44 a^{3} + 23 a^{2} - 28 a - 10\) , \( 3 a^{5} - a^{4} - 21 a^{3} - 8 a^{2} + 12 a + 3\) , \( -5 a^{5} + a^{4} + 37 a^{3} + 18 a^{2} - 26 a - 7\) , \( -82 a^{5} + 24 a^{4} + 591 a^{3} + 257 a^{2} - 391 a - 116\) , \( 161 a^{5} - 45 a^{4} - 1160 a^{3} - 509 a^{2} + 769 a + 227\bigr] \) |
${y}^2+\left(-6a^{5}+a^{4}+44a^{3}+23a^{2}-28a-10\right){x}{y}+\left(-5a^{5}+a^{4}+37a^{3}+18a^{2}-26a-7\right){y}={x}^{3}+\left(3a^{5}-a^{4}-21a^{3}-8a^{2}+12a+3\right){x}^{2}+\left(-82a^{5}+24a^{4}+591a^{3}+257a^{2}-391a-116\right){x}+161a^{5}-45a^{4}-1160a^{3}-509a^{2}+769a+227$ |
| 71.3-b2 |
71.3-b |
$2$ |
$2$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.3 |
\( 71 \) |
\( -71 \) |
$69.83308$ |
$(-a^5+a^4+7a^3-2a^2-7a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.003038782$ |
$267854.1885$ |
2.22863 |
\( \frac{1827756}{71} a^{5} - \frac{703934}{71} a^{4} - \frac{12966589}{71} a^{3} - \frac{4193644}{71} a^{2} + 114206 a + \frac{775598}{71} \) |
\( \bigl[-5 a^{5} + a^{4} + 37 a^{3} + 18 a^{2} - 26 a - 7\) , \( 3 a^{5} - a^{4} - 21 a^{3} - 9 a^{2} + 13 a + 3\) , \( 3 a^{5} - a^{4} - 21 a^{3} - 9 a^{2} + 13 a + 4\) , \( 13 a^{5} - 3 a^{4} - 93 a^{3} - 43 a^{2} + 59 a + 17\) , \( -7 a^{5} + 2 a^{4} + 52 a^{3} + 25 a^{2} - 31 a - 9\bigr] \) |
${y}^2+\left(-5a^{5}+a^{4}+37a^{3}+18a^{2}-26a-7\right){x}{y}+\left(3a^{5}-a^{4}-21a^{3}-9a^{2}+13a+4\right){y}={x}^{3}+\left(3a^{5}-a^{4}-21a^{3}-9a^{2}+13a+3\right){x}^{2}+\left(13a^{5}-3a^{4}-93a^{3}-43a^{2}+59a+17\right){x}-7a^{5}+2a^{4}+52a^{3}+25a^{2}-31a-9$ |
| 71.3-c1 |
71.3-c |
$4$ |
$6$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.3 |
\( 71 \) |
\( - 71^{6} \) |
$69.83308$ |
$(-a^5+a^4+7a^3-2a^2-7a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$81$ |
\( 2 \) |
$1$ |
$17.45399107$ |
1.29032 |
\( -\frac{12188349191966910976545591421520911263}{128100283921} a^{5} + \frac{2795083691338226342388959981793334127}{128100283921} a^{4} + \frac{87472547643214485589591447553049289390}{128100283921} a^{3} + \frac{43036275357468336314681380908411365576}{128100283921} a^{2} - \frac{734527173722240035758902620347146720}{1804229351} a - \frac{15815145011644535445549863476994175471}{128100283921} \) |
\( \bigl[-6 a^{5} + a^{4} + 44 a^{3} + 23 a^{2} - 28 a - 9\) , \( 3 a^{5} - a^{4} - 21 a^{3} - 10 a^{2} + 12 a + 6\) , \( a^{5} - 7 a^{3} - 5 a^{2} + 3 a + 2\) , \( 301 a^{5} - 10 a^{4} - 2223 a^{3} - 1411 a^{2} + 1326 a + 389\) , \( 2885 a^{5} - 180 a^{4} - 21196 a^{3} - 12957 a^{2} + 12811 a + 3947\bigr] \) |
${y}^2+\left(-6a^{5}+a^{4}+44a^{3}+23a^{2}-28a-9\right){x}{y}+\left(a^{5}-7a^{3}-5a^{2}+3a+2\right){y}={x}^{3}+\left(3a^{5}-a^{4}-21a^{3}-10a^{2}+12a+6\right){x}^{2}+\left(301a^{5}-10a^{4}-2223a^{3}-1411a^{2}+1326a+389\right){x}+2885a^{5}-180a^{4}-21196a^{3}-12957a^{2}+12811a+3947$ |
| 71.3-c2 |
71.3-c |
$4$ |
$6$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.3 |
\( 71 \) |
\( -71 \) |
$69.83308$ |
$(-a^5+a^4+7a^3-2a^2-7a)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$25447.91898$ |
1.29032 |
\( \frac{8686953}{71} a^{5} + \frac{11816056}{71} a^{4} - \frac{25529599}{71} a^{3} - \frac{21656895}{71} a^{2} + 210469 a + \frac{5162701}{71} \) |
\( \bigl[-6 a^{5} + 2 a^{4} + 43 a^{3} + 17 a^{2} - 28 a - 6\) , \( -a^{2} + a + 3\) , \( a^{5} - a^{4} - 6 a^{3} + a^{2} + 3 a - 1\) , \( 8 a^{5} - a^{4} - 59 a^{3} - 34 a^{2} + 38 a + 18\) , \( 20 a^{5} - 5 a^{4} - 144 a^{3} - 69 a^{2} + 91 a + 29\bigr] \) |
${y}^2+\left(-6a^{5}+2a^{4}+43a^{3}+17a^{2}-28a-6\right){x}{y}+\left(a^{5}-a^{4}-6a^{3}+a^{2}+3a-1\right){y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(8a^{5}-a^{4}-59a^{3}-34a^{2}+38a+18\right){x}+20a^{5}-5a^{4}-144a^{3}-69a^{2}+91a+29$ |
| 71.3-c3 |
71.3-c |
$4$ |
$6$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.3 |
\( 71 \) |
\( - 71^{2} \) |
$69.83308$ |
$(-a^5+a^4+7a^3-2a^2-7a)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$12723.95949$ |
1.29032 |
\( \frac{12287267278054020}{5041} a^{5} + \frac{23544415398400945}{5041} a^{4} - \frac{17366454259785204}{5041} a^{3} - \frac{26109054624611151}{5041} a^{2} + \frac{138912635005304}{71} a + \frac{4211222685079557}{5041} \) |
\( \bigl[-6 a^{5} + 2 a^{4} + 43 a^{3} + 17 a^{2} - 28 a - 6\) , \( -a^{2} + a + 3\) , \( a^{5} - a^{4} - 6 a^{3} + a^{2} + 3 a - 1\) , \( 48 a^{5} - 11 a^{4} - 344 a^{3} - 169 a^{2} + 198 a + 53\) , \( 315 a^{5} - 72 a^{4} - 2261 a^{3} - 1115 a^{2} + 1350 a + 413\bigr] \) |
${y}^2+\left(-6a^{5}+2a^{4}+43a^{3}+17a^{2}-28a-6\right){x}{y}+\left(a^{5}-a^{4}-6a^{3}+a^{2}+3a-1\right){y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(48a^{5}-11a^{4}-344a^{3}-169a^{2}+198a+53\right){x}+315a^{5}-72a^{4}-2261a^{3}-1115a^{2}+1350a+413$ |
| 71.3-c4 |
71.3-c |
$4$ |
$6$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.3 |
\( 71 \) |
\( - 71^{3} \) |
$69.83308$ |
$(-a^5+a^4+7a^3-2a^2-7a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$81$ |
\( 1 \) |
$1$ |
$34.90798215$ |
1.29032 |
\( \frac{4417193206959849167}{357911} a^{5} - \frac{721150030079876042}{357911} a^{4} - \frac{32357369288619010195}{357911} a^{3} - \frac{16366003801804012761}{357911} a^{2} + \frac{273597199434573362}{5041} a + \frac{5918888612952222917}{357911} \) |
\( \bigl[-6 a^{5} + a^{4} + 44 a^{3} + 23 a^{2} - 28 a - 9\) , \( 3 a^{5} - a^{4} - 21 a^{3} - 10 a^{2} + 12 a + 6\) , \( a^{5} - 7 a^{3} - 5 a^{2} + 3 a + 2\) , \( -4 a^{5} + 55 a^{4} - 43 a^{3} - 296 a^{2} + 126 a + 59\) , \( -181 a^{5} + 461 a^{4} + 739 a^{3} - 1746 a^{2} + 376 a + 244\bigr] \) |
${y}^2+\left(-6a^{5}+a^{4}+44a^{3}+23a^{2}-28a-9\right){x}{y}+\left(a^{5}-7a^{3}-5a^{2}+3a+2\right){y}={x}^{3}+\left(3a^{5}-a^{4}-21a^{3}-10a^{2}+12a+6\right){x}^{2}+\left(-4a^{5}+55a^{4}-43a^{3}-296a^{2}+126a+59\right){x}-181a^{5}+461a^{4}+739a^{3}-1746a^{2}+376a+244$ |
*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.
