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Label Class Base field Conductor norm Rank Torsion CM Sato-Tate Regulator Period Leading coeff j-invariant Weierstrass coefficients Weierstrass equation
71.5-a1 71.5-a 6.6.300125.1 \( 71 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $2697.010303$ 1.23075 \( \frac{18671790400300000}{1804229351} a^{5} - \frac{5223136808131864}{1804229351} a^{4} - \frac{134463355085824784}{1804229351} a^{3} - \frac{59498874802185092}{1804229351} a^{2} + \frac{87859926083703700}{1804229351} a + \frac{25940020939985535}{1804229351} \) \( \bigl[-6 a^{5} + 2 a^{4} + 43 a^{3} + 17 a^{2} - 29 a - 7\) , \( a^{5} - 8 a^{3} - 4 a^{2} + 7 a + 2\) , \( -6 a^{5} + a^{4} + 44 a^{3} + 23 a^{2} - 28 a - 10\) , \( 108 a^{5} - 25 a^{4} - 776 a^{3} - 378 a^{2} + 466 a + 140\) , \( 536 a^{5} - 123 a^{4} - 3847 a^{3} - 1892 a^{2} + 2295 a + 696\bigr] \) ${y}^2+\left(-6a^{5}+2a^{4}+43a^{3}+17a^{2}-29a-7\right){x}{y}+\left(-6a^{5}+a^{4}+44a^{3}+23a^{2}-28a-10\right){y}={x}^{3}+\left(a^{5}-8a^{3}-4a^{2}+7a+2\right){x}^{2}+\left(108a^{5}-25a^{4}-776a^{3}-378a^{2}+466a+140\right){x}+536a^{5}-123a^{4}-3847a^{3}-1892a^{2}+2295a+696$
71.5-a2 71.5-a 6.6.300125.1 \( 71 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $1348.505151$ 1.23075 \( -\frac{41014241389245672911233616}{3255243551009881201} a^{5} + \frac{31251665362953723702641564}{3255243551009881201} a^{4} + \frac{283323882933070565416889280}{3255243551009881201} a^{3} - \frac{8921536262849968349901544}{3255243551009881201} a^{2} - \frac{210325636400488563887402742}{3255243551009881201} a + \frac{49837787027906807246579723}{3255243551009881201} \) \( \bigl[-8 a^{5} + 2 a^{4} + 58 a^{3} + 27 a^{2} - 38 a - 12\) , \( -6 a^{5} + 2 a^{4} + 43 a^{3} + 17 a^{2} - 30 a - 6\) , \( -3 a^{5} + 23 a^{3} + 14 a^{2} - 17 a - 6\) , \( 10 a^{5} - 2 a^{4} - 72 a^{3} - 41 a^{2} + 36 a + 12\) , \( 15 a^{5} - 27 a^{4} - 174 a^{3} - 76 a^{2} + 101 a + 30\bigr] \) ${y}^2+\left(-8a^{5}+2a^{4}+58a^{3}+27a^{2}-38a-12\right){x}{y}+\left(-3a^{5}+23a^{3}+14a^{2}-17a-6\right){y}={x}^{3}+\left(-6a^{5}+2a^{4}+43a^{3}+17a^{2}-30a-6\right){x}^{2}+\left(10a^{5}-2a^{4}-72a^{3}-41a^{2}+36a+12\right){x}+15a^{5}-27a^{4}-174a^{3}-76a^{2}+101a+30$
71.5-b1 71.5-b 6.6.300125.1 \( 71 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $0.006077565$ $66963.54714$ 2.22863 \( -\frac{3949307338048}{5041} a^{5} + \frac{657783939680}{5041} a^{4} + \frac{28702201019885}{5041} a^{3} + \frac{15411298823022}{5041} a^{2} - \frac{17627754558666}{5041} a - \frac{5427054812636}{5041} \) \( \bigl[-a^{5} + 8 a^{3} + 5 a^{2} - 7 a - 2\) , \( -6 a^{5} + 2 a^{4} + 43 a^{3} + 17 a^{2} - 28 a - 8\) , \( -3 a^{5} + 23 a^{3} + 14 a^{2} - 17 a - 5\) , \( 10 a^{5} - 2 a^{4} - 69 a^{3} - 40 a^{2} + 49 a + 14\) , \( 12 a^{5} + 4 a^{4} - 100 a^{3} - 51 a^{2} + 68 a + 20\bigr] \) ${y}^2+\left(-a^{5}+8a^{3}+5a^{2}-7a-2\right){x}{y}+\left(-3a^{5}+23a^{3}+14a^{2}-17a-5\right){y}={x}^{3}+\left(-6a^{5}+2a^{4}+43a^{3}+17a^{2}-28a-8\right){x}^{2}+\left(10a^{5}-2a^{4}-69a^{3}-40a^{2}+49a+14\right){x}+12a^{5}+4a^{4}-100a^{3}-51a^{2}+68a+20$
71.5-b2 71.5-b 6.6.300125.1 \( 71 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $0.003038782$ $267854.1885$ 2.22863 \( \frac{5286451}{71} a^{5} - \frac{1105287}{71} a^{4} - \frac{38268478}{71} a^{3} - \frac{19161741}{71} a^{2} + \frac{24321390}{71} a + \frac{7597551}{71} \) \( \bigl[a^{5} - 7 a^{3} - 5 a^{2} + 2 a + 2\) , \( a^{5} - 7 a^{3} - 5 a^{2} + 3 a + 1\) , \( -3 a^{5} + 23 a^{3} + 14 a^{2} - 17 a - 5\) , \( -3 a^{5} + 2 a^{4} + 20 a^{3} + 4 a^{2} - 10 a - 4\) , \( -2 a^{5} + a^{4} + 14 a^{3} + 4 a^{2} - 8 a - 2\bigr] \) ${y}^2+\left(a^{5}-7a^{3}-5a^{2}+2a+2\right){x}{y}+\left(-3a^{5}+23a^{3}+14a^{2}-17a-5\right){y}={x}^{3}+\left(a^{5}-7a^{3}-5a^{2}+3a+1\right){x}^{2}+\left(-3a^{5}+2a^{4}+20a^{3}+4a^{2}-10a-4\right){x}-2a^{5}+a^{4}+14a^{3}+4a^{2}-8a-2$
71.5-c1 71.5-c 6.6.300125.1 \( 71 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $17.45399107$ 1.29032 \( \frac{688385227545894441783394700691160040}{128100283921} a^{5} - \frac{1455011470820246681274446975813913565}{128100283921} a^{4} - \frac{3198310313219337512071267632384924449}{128100283921} a^{3} + \frac{4938596772879076842480778038133182698}{128100283921} a^{2} - \frac{681214996706145908023368640065836270}{128100283921} a - \frac{618129402248792677846156102081913073}{128100283921} \) \( \bigl[-8 a^{5} + 2 a^{4} + 58 a^{3} + 27 a^{2} - 39 a - 12\) , \( -4 a^{5} + a^{4} + 29 a^{3} + 14 a^{2} - 21 a - 6\) , \( 1\) , \( -944 a^{5} + 1869 a^{4} + 4540 a^{3} - 6026 a^{2} + 652 a + 593\) , \( -34009 a^{5} + 72266 a^{4} + 157382 a^{3} - 246003 a^{2} + 35333 a + 30593\bigr] \) ${y}^2+\left(-8a^{5}+2a^{4}+58a^{3}+27a^{2}-39a-12\right){x}{y}+{y}={x}^{3}+\left(-4a^{5}+a^{4}+29a^{3}+14a^{2}-21a-6\right){x}^{2}+\left(-944a^{5}+1869a^{4}+4540a^{3}-6026a^{2}+652a+593\right){x}-34009a^{5}+72266a^{4}+157382a^{3}-246003a^{2}+35333a+30593$
71.5-c2 71.5-c 6.6.300125.1 \( 71 \) 0 $\Z/6\Z$ $\mathrm{SU}(2)$ $1$ $25447.91898$ 1.29032 \( \frac{3453103834}{71} a^{5} - \frac{966256218}{71} a^{4} - \frac{24868045898}{71} a^{3} - \frac{11002493289}{71} a^{2} + \frac{16250723867}{71} a + \frac{4795868035}{71} \) \( \bigl[-2 a^{5} + a^{4} + 14 a^{3} + 4 a^{2} - 10 a - 1\) , \( -7 a^{5} + a^{4} + 51 a^{3} + 28 a^{2} - 32 a - 13\) , \( -a^{5} + 8 a^{3} + 4 a^{2} - 6 a - 1\) , \( 5 a^{5} - 38 a^{3} - 22 a^{2} + 26 a + 9\) , \( -3 a^{5} + a^{4} + 21 a^{3} + 8 a^{2} - 12 a - 4\bigr] \) ${y}^2+\left(-2a^{5}+a^{4}+14a^{3}+4a^{2}-10a-1\right){x}{y}+\left(-a^{5}+8a^{3}+4a^{2}-6a-1\right){y}={x}^{3}+\left(-7a^{5}+a^{4}+51a^{3}+28a^{2}-32a-13\right){x}^{2}+\left(5a^{5}-38a^{3}-22a^{2}+26a+9\right){x}-3a^{5}+a^{4}+21a^{3}+8a^{2}-12a-4$
71.5-c3 71.5-c 6.6.300125.1 \( 71 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $34.90798215$ 1.29032 \( -\frac{235353321428599978}{357911} a^{5} + \frac{580741577725442818}{357911} a^{4} + \frac{1162430939297012415}{357911} a^{3} - \frac{1879545228122345414}{357911} a^{2} + \frac{264656034786687147}{357911} a + \frac{239407872830503392}{357911} \) \( \bigl[a^{5} - a^{4} - 6 a^{3} + a^{2} + 3 a\) , \( -4 a^{5} + 30 a^{3} + 19 a^{2} - 18 a - 9\) , \( -6 a^{5} + 2 a^{4} + 43 a^{3} + 17 a^{2} - 29 a - 6\) , \( -36 a^{5} - 3 a^{4} + 278 a^{3} + 176 a^{2} - 156 a - 75\) , \( 1119 a^{5} - 361 a^{4} - 7802 a^{3} - 3654 a^{2} + 4623 a + 1358\bigr] \) ${y}^2+\left(a^{5}-a^{4}-6a^{3}+a^{2}+3a\right){x}{y}+\left(-6a^{5}+2a^{4}+43a^{3}+17a^{2}-29a-6\right){y}={x}^{3}+\left(-4a^{5}+30a^{3}+19a^{2}-18a-9\right){x}^{2}+\left(-36a^{5}-3a^{4}+278a^{3}+176a^{2}-156a-75\right){x}+1119a^{5}-361a^{4}-7802a^{3}-3654a^{2}+4623a+1358$
71.5-c4 71.5-c 6.6.300125.1 \( 71 \) 0 $\Z/6\Z$ $\mathrm{SU}(2)$ $1$ $12723.95949$ 1.29032 \( \frac{6615986088905408509}{5041} a^{5} - \frac{1851862023389315072}{5041} a^{4} - \frac{47645414547979919968}{5041} a^{3} - \frac{21077149500839791045}{5041} a^{2} + \frac{31134395160454919709}{5041} a + \frac{9187684081459759389}{5041} \) \( \bigl[-2 a^{5} + 15 a^{3} + 10 a^{2} - 9 a - 5\) , \( a^{5} - 7 a^{3} - 5 a^{2} + a + 3\) , \( a + 1\) , \( 1534 a^{5} - 351 a^{4} - 11009 a^{3} - 5421 a^{2} + 6557 a + 1992\) , \( -18999 a^{5} + 4356 a^{4} + 136350 a^{3} + 67090 a^{2} - 81287 a - 24651\bigr] \) ${y}^2+\left(-2a^{5}+15a^{3}+10a^{2}-9a-5\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{5}-7a^{3}-5a^{2}+a+3\right){x}^{2}+\left(1534a^{5}-351a^{4}-11009a^{3}-5421a^{2}+6557a+1992\right){x}-18999a^{5}+4356a^{4}+136350a^{3}+67090a^{2}-81287a-24651$
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  *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.