Learn more

Refine search


Results (1-50 of 1262 matches)

Next   Download to        
Label Class Base field Conductor norm Rank Torsion CM Regulator Period Leading coeff j-invariant Weierstrass coefficients Weierstrass equation
8.1-a1 8.1-a \(\Q(\zeta_{9})^+\) \( 2^{3} \) $0$ $\Z/21\Z$ $1$ $386.8008344$ 0.292366465 \( -\frac{140625}{8} \) \( \bigl[a^{2} + a - 1\) , \( -a^{2} + a + 2\) , \( a^{2} - 2\) , \( 13 a^{2} + 2 a - 44\) , \( -22 a^{2} - 3 a + 88\bigr] \) ${y}^2+\left(a^{2}+a-1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(13a^{2}+2a-44\right){x}-22a^{2}-3a+88$
8.1-a2 8.1-a \(\Q(\zeta_{9})^+\) \( 2^{3} \) $0$ $\Z/3\Z$ $1$ $1.127699225$ 0.292366465 \( -\frac{1159088625}{2097152} \) \( \bigl[a\) , \( -1\) , \( a^{2} - 1\) , \( 125040 a^{2} - 43189 a - 360486\) , \( 56242078 a^{2} - 19527401 a - 161952511\bigr] \) ${y}^2+a{x}{y}+\left(a^{2}-1\right){y}={x}^{3}-{x}^{2}+\left(125040a^{2}-43189a-360486\right){x}+56242078a^{2}-19527401a-161952511$
8.1-a3 8.1-a \(\Q(\zeta_{9})^+\) \( 2^{3} \) $0$ $\Z/7\Z$ $1$ $128.9336114$ 0.292366465 \( \frac{3375}{2} \) \( \bigl[1\) , \( a^{2} + a - 3\) , \( a^{2} + a - 1\) , \( -a^{2} + 4\) , \( -a - 2\bigr] \) ${y}^2+{x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+\left(a^{2}+a-3\right){x}^{2}+\left(-a^{2}+4\right){x}-a-2$
8.1-a4 8.1-a \(\Q(\zeta_{9})^+\) \( 2^{3} \) $0$ $\mathsf{trivial}$ $1$ $0.375899741$ 0.292366465 \( -\frac{189613868625}{128} \) \( \bigl[a + 1\) , \( -a\) , \( a^{2} - 1\) , \( 17115 a^{2} - 5866 a - 49431\) , \( 1410206 a^{2} - 489214 a - 4061553\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}-a{x}^{2}+\left(17115a^{2}-5866a-49431\right){x}+1410206a^{2}-489214a-4061553$
17.1-a1 17.1-a \(\Q(\zeta_{9})^+\) \( 17 \) $0$ $\Z/2\Z$ $1$ $18.26148809$ 0.507263558 \( -\frac{7715927403056521557}{17} a^{2} + \frac{2679713465103470916}{17} a + \frac{22217127489402511122}{17} \) \( \bigl[a^{2} - 2\) , \( -a^{2} + 3\) , \( 0\) , \( 99 a^{2} - 64 a - 331\) , \( 958 a^{2} - 218 a - 2583\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(99a^{2}-64a-331\right){x}+958a^{2}-218a-2583$
17.1-a2 17.1-a \(\Q(\zeta_{9})^+\) \( 17 \) $0$ $\Z/2\Z$ $1$ $4.565372023$ 0.507263558 \( \frac{261445487080230981}{83521} a^{2} - \frac{400557711948933636}{83521} a - \frac{170646420428250786}{83521} \) \( \bigl[a^{2} - 2\) , \( -a^{2} + 3\) , \( 0\) , \( -11 a^{2} - 24 a - 21\) , \( -38 a^{2} - 104 a - 85\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(-11a^{2}-24a-21\right){x}-38a^{2}-104a-85$
17.1-a3 17.1-a \(\Q(\zeta_{9})^+\) \( 17 \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $36.52297618$ 0.507263558 \( -\frac{119617489449}{289} a^{2} + \frac{41241472785}{289} a + \frac{344991489660}{289} \) \( \bigl[a^{2} - 2\) , \( -a^{2} + 3\) , \( 0\) , \( 4 a^{2} - 4 a - 16\) , \( 20 a^{2} - 7 a - 58\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(4a^{2}-4a-16\right){x}+20a^{2}-7a-58$
17.1-a4 17.1-a \(\Q(\zeta_{9})^+\) \( 17 \) $0$ $\Z/4\Z$ $1$ $73.04595237$ 0.507263558 \( -\frac{219348}{17} a^{2} + \frac{90963}{17} a + \frac{633150}{17} \) \( \bigl[a^{2} - 2\) , \( -a^{2} + 3\) , \( 0\) , \( -a^{2} + a + 4\) , \( 0\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(-a^{2}+a+4\right){x}$
17.1-a5 17.1-a \(\Q(\zeta_{9})^+\) \( 17 \) $0$ $\Z/6\Z$ $1$ $54.78446428$ 0.507263558 \( \frac{11224223245743811083}{4913} a^{2} + \frac{21094639493182324371}{4913} a + \frac{5972284442955277116}{4913} \) \( \bigl[1\) , \( a^{2} - 2\) , \( a^{2} + a - 2\) , \( 1029 a^{2} - 331 a - 3016\) , \( 20866 a^{2} - 7345 a - 59899\bigr] \) ${y}^2+{x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(1029a^{2}-331a-3016\right){x}+20866a^{2}-7345a-59899$
17.1-a6 17.1-a \(\Q(\zeta_{9})^+\) \( 17 \) $0$ $\Z/2\Z\oplus\Z/6\Z$ $1$ $109.5689285$ 0.507263558 \( \frac{418603459872780}{24137569} a^{2} + \frac{786688502074839}{24137569} a + \frac{222772245824754}{24137569} \) \( \bigl[1\) , \( a^{2} - 2\) , \( a^{2} + a - 2\) , \( 79 a^{2} - 26 a - 231\) , \( 122 a^{2} - 44 a - 351\bigr] \) ${y}^2+{x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(79a^{2}-26a-231\right){x}+122a^{2}-44a-351$
17.1-a7 17.1-a \(\Q(\zeta_{9})^+\) \( 17 \) $0$ $\Z/12\Z$ $1$ $219.1378571$ 0.507263558 \( -\frac{170514783}{4913} a^{2} + \frac{42561684}{4913} a + \frac{469465740}{4913} \) \( \bigl[1\) , \( a^{2} - 2\) , \( a^{2} + a - 2\) , \( 44 a^{2} - 16 a - 126\) , \( -207 a^{2} + 71 a + 595\bigr] \) ${y}^2+{x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(44a^{2}-16a-126\right){x}-207a^{2}+71a+595$
17.1-a8 17.1-a \(\Q(\zeta_{9})^+\) \( 17 \) $0$ $\Z/6\Z$ $1$ $13.69611607$ 0.507263558 \( \frac{4075073731548124101}{582622237229761} a^{2} - \frac{3433122709460467011}{582622237229761} a - \frac{810676269288190764}{582622237229761} \) \( \bigl[1\) , \( a^{2} - 2\) , \( a^{2} + a - 2\) , \( -311 a^{2} + 119 a + 874\) , \( 1414 a^{2} - 463 a - 4127\bigr] \) ${y}^2+{x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(-311a^{2}+119a+874\right){x}+1414a^{2}-463a-4127$
17.2-a1 17.2-a \(\Q(\zeta_{9})^+\) \( 17 \) $0$ $\Z/12\Z$ $1$ $219.1378571$ 0.507263558 \( \frac{42561684}{4913} a^{2} + \frac{127953099}{4913} a + \frac{43312806}{4913} \) \( \bigl[a^{2} - 1\) , \( -a + 1\) , \( a^{2} + a - 2\) , \( -30 a^{2} + 43 a + 21\) , \( 103 a^{2} - 160 a - 66\bigr] \) ${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-30a^{2}+43a+21\right){x}+103a^{2}-160a-66$
17.2-a2 17.2-a \(\Q(\zeta_{9})^+\) \( 17 \) $0$ $\Z/6\Z$ $1$ $13.69611607$ 0.507263558 \( -\frac{3433122709460467011}{582622237229761} a^{2} - \frac{641951022087657090}{582622237229761} a + \frac{14205716612728991460}{582622237229761} \) \( \bigl[a^{2} - 1\) , \( -a + 1\) , \( a^{2} + a - 2\) , \( -455 a^{2} + 738 a + 216\) , \( 7473 a^{2} - 11274 a - 5210\bigr] \) ${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-455a^{2}+738a+216\right){x}+7473a^{2}-11274a-5210$
17.2-a3 17.2-a \(\Q(\zeta_{9})^+\) \( 17 \) $0$ $\Z/2\Z\oplus\Z/6\Z$ $1$ $109.5689285$ 0.507263558 \( \frac{786688502074839}{24137569} a^{2} - \frac{1205291961947619}{24137569} a - \frac{513397838579364}{24137569} \) \( \bigl[a^{2} - 1\) , \( -a + 1\) , \( a^{2} + a - 2\) , \( -475 a^{2} + 733 a + 296\) , \( 7691 a^{2} - 11788 a - 5014\bigr] \) ${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-475a^{2}+733a+296\right){x}+7691a^{2}-11788a-5014$
17.2-a4 17.2-a \(\Q(\zeta_{9})^+\) \( 17 \) $0$ $\Z/6\Z$ $1$ $54.78446428$ 0.507263558 \( \frac{21094639493182324371}{4913} a^{2} - \frac{32318862738926135454}{4913} a - \frac{13768548051921749460}{4913} \) \( \bigl[a^{2} - 1\) , \( -a + 1\) , \( a^{2} + a - 2\) , \( -7615 a^{2} + 11768 a + 4776\) , \( 514961 a^{2} - 789514 a - 335090\bigr] \) ${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-7615a^{2}+11768a+4776\right){x}+514961a^{2}-789514a-335090$
17.2-a5 17.2-a \(\Q(\zeta_{9})^+\) \( 17 \) $0$ $\Z/2\Z$ $1$ $18.26148809$ 0.507263558 \( \frac{2679713465103470916}{17} a^{2} + \frac{5036213937953050641}{17} a + \frac{1425845753082526176}{17} \) \( \bigl[a\) , \( a + 1\) , \( 0\) , \( -64 a^{2} - 35 a - 5\) , \( -218 a^{2} - 740 a - 231\bigr] \) ${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-64a^{2}-35a-5\right){x}-218a^{2}-740a-231$
17.2-a6 17.2-a \(\Q(\zeta_{9})^+\) \( 17 \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $36.52297618$ 0.507263558 \( \frac{41241472785}{289} a^{2} + \frac{78376016664}{289} a + \frac{23273565192}{289} \) \( \bigl[a\) , \( a + 1\) , \( 0\) , \( -4 a^{2}\) , \( -7 a^{2} - 13 a - 4\bigr] \) ${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}-4a^{2}{x}-7a^{2}-13a-4$
17.2-a7 17.2-a \(\Q(\zeta_{9})^+\) \( 17 \) $0$ $\Z/4\Z$ $1$ $73.04595237$ 0.507263558 \( \frac{90963}{17} a^{2} + \frac{128385}{17} a + \frac{12528}{17} \) \( \bigl[a\) , \( a + 1\) , \( 0\) , \( a^{2}\) , \( 0\bigr] \) ${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+a^{2}{x}$
17.2-a8 17.2-a \(\Q(\zeta_{9})^+\) \( 17 \) $0$ $\Z/2\Z$ $1$ $4.565372023$ 0.507263558 \( -\frac{400557711948933636}{83521} a^{2} + \frac{139112224868702655}{83521} a + \frac{1153359977630078448}{83521} \) \( \bigl[a\) , \( a + 1\) , \( 0\) , \( -24 a^{2} + 35 a + 5\) , \( -104 a^{2} + 142 a + 47\bigr] \) ${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-24a^{2}+35a+5\right){x}-104a^{2}+142a+47$
17.3-a1 17.3-a \(\Q(\zeta_{9})^+\) \( 17 \) $0$ $\Z/6\Z$ $1$ $54.78446428$ 0.507263558 \( -\frac{32318862738926135454}{4913} a^{2} + \frac{11224223245743811083}{4913} a + \frac{93058456412295170190}{4913} \) \( \bigl[a^{2} - 2\) , \( -a^{2} - a + 1\) , \( a\) , \( 28433 a^{2} - 9539 a - 82497\) , \( -3025589 a^{2} + 1055150 a + 8703613\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}-a+1\right){x}^{2}+\left(28433a^{2}-9539a-82497\right){x}-3025589a^{2}+1055150a+8703613$
17.3-a2 17.3-a \(\Q(\zeta_{9})^+\) \( 17 \) $0$ $\Z/6\Z$ $1$ $13.69611607$ 0.507263558 \( -\frac{641951022087657090}{582622237229761} a^{2} + \frac{4075073731548124101}{582622237229761} a + \frac{8623373237983371618}{582622237229761} \) \( \bigl[a^{2} - 2\) , \( -a^{2} - a + 1\) , \( a\) , \( 1913 a^{2} - 759 a - 5327\) , \( -44365 a^{2} + 15660 a + 127267\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}-a+1\right){x}^{2}+\left(1913a^{2}-759a-5327\right){x}-44365a^{2}+15660a+127267$
17.3-a3 17.3-a \(\Q(\zeta_{9})^+\) \( 17 \) $0$ $\Z/2\Z\oplus\Z/6\Z$ $1$ $109.5689285$ 0.507263558 \( -\frac{1205291961947619}{24137569} a^{2} + \frac{418603459872780}{24137569} a + \frac{3470563089465552}{24137569} \) \( \bigl[a^{2} - 2\) , \( -a^{2} - a + 1\) , \( a\) , \( 1773 a^{2} - 589 a - 5152\) , \( -46301 a^{2} + 16115 a + 133250\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}-a+1\right){x}^{2}+\left(1773a^{2}-589a-5152\right){x}-46301a^{2}+16115a+133250$
17.3-a4 17.3-a \(\Q(\zeta_{9})^+\) \( 17 \) $0$ $\Z/12\Z$ $1$ $219.1378571$ 0.507263558 \( \frac{127953099}{4913} a^{2} - \frac{170514783}{4913} a - \frac{127470024}{4913} \) \( \bigl[a^{2} - 2\) , \( -a^{2} - a + 1\) , \( a\) , \( 98 a^{2} - 19 a - 307\) , \( -668 a^{2} + 195 a + 1990\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}-a+1\right){x}^{2}+\left(98a^{2}-19a-307\right){x}-668a^{2}+195a+1990$
17.3-a5 17.3-a \(\Q(\zeta_{9})^+\) \( 17 \) $0$ $\Z/2\Z$ $1$ $4.565372023$ 0.507263558 \( \frac{139112224868702655}{83521} a^{2} + \frac{261445487080230981}{83521} a + \frac{74020103994805866}{83521} \) \( \bigl[a^{2} + a - 2\) , \( a^{2} + a - 2\) , \( a^{2} + a - 1\) , \( 35 a^{2} - 8 a - 113\) , \( 133 a^{2} - 47 a - 411\bigr] \) ${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+\left(a^{2}+a-2\right){x}^{2}+\left(35a^{2}-8a-113\right){x}+133a^{2}-47a-411$
17.3-a6 17.3-a \(\Q(\zeta_{9})^+\) \( 17 \) $0$ $\Z/2\Z$ $1$ $18.26148809$ 0.507263558 \( \frac{5036213937953050641}{17} a^{2} - \frac{7715927403056521557}{17} a - \frac{3287155192616633274}{17} \) \( \bigl[a^{2} + a - 2\) , \( a^{2} + a - 2\) , \( a^{2} + a - 1\) , \( -35 a^{2} + 102 a - 63\) , \( -639 a^{2} + 789 a + 777\bigr] \) ${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+\left(a^{2}+a-2\right){x}^{2}+\left(-35a^{2}+102a-63\right){x}-639a^{2}+789a+777$
17.3-a7 17.3-a \(\Q(\zeta_{9})^+\) \( 17 \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $36.52297618$ 0.507263558 \( \frac{78376016664}{289} a^{2} - \frac{119617489449}{289} a - \frac{50995522566}{289} \) \( \bigl[a^{2} + a - 2\) , \( a^{2} + a - 2\) , \( a^{2} + a - 1\) , \( 7 a - 8\) , \( -7 a^{2} + 11 a + 7\bigr] \) ${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+\left(a^{2}+a-2\right){x}^{2}+\left(7a-8\right){x}-7a^{2}+11a+7$
17.3-a8 17.3-a \(\Q(\zeta_{9})^+\) \( 17 \) $0$ $\Z/4\Z$ $1$ $73.04595237$ 0.507263558 \( \frac{128385}{17} a^{2} - \frac{219348}{17} a - \frac{62316}{17} \) \( \bigl[a^{2} + a - 2\) , \( a^{2} + a - 2\) , \( a^{2} + a - 1\) , \( 2 a + 2\) , \( a^{2} + a - 1\bigr] \) ${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+\left(a^{2}+a-2\right){x}^{2}+\left(2a+2\right){x}+a^{2}+a-1$
27.1-a1 27.1-a \(\Q(\zeta_{9})^+\) \( 3^{3} \) $0$ $\mathsf{trivial}$ $-27$ $1$ $1.837900525$ 0.612633508 \( -12288000 \) \( \bigl[0\) , \( a^{2} + a - 2\) , \( a^{2} - 1\) , \( 280 a^{2} - 89 a - 818\) , \( 2805 a^{2} - 963 a - 8098\bigr] \) ${y}^2+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{2}+a-2\right){x}^{2}+\left(280a^{2}-89a-818\right){x}+2805a^{2}-963a-8098$
27.1-a2 27.1-a \(\Q(\zeta_{9})^+\) \( 3^{3} \) $0$ $\Z/3\Z$ $-3$ $1$ $49.62331419$ 0.612633508 \( 0 \) \( \bigl[0\) , \( a^{2} + a - 2\) , \( a^{2} - 1\) , \( a + 2\) , \( 12 a^{2} - 4 a - 35\bigr] \) ${y}^2+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{2}+a-2\right){x}^{2}+\left(a+2\right){x}+12a^{2}-4a-35$
27.1-a3 27.1-a \(\Q(\zeta_{9})^+\) \( 3^{3} \) $0$ $\Z/9\Z$ $-27$ $1$ $446.6098277$ 0.612633508 \( -12288000 \) \( \bigl[0\) , \( a^{2} - 1\) , \( a + 1\) , \( 17 a^{2} - 3 a - 53\) , \( -56 a^{2} + 17 a + 164\bigr] \) ${y}^2+\left(a+1\right){y}={x}^{3}+\left(a^{2}-1\right){x}^{2}+\left(17a^{2}-3a-53\right){x}-56a^{2}+17a+164$
27.1-a4 27.1-a \(\Q(\zeta_{9})^+\) \( 3^{3} \) $0$ $\Z/9\Z$ $-3$ $1$ $148.8699425$ 0.612633508 \( 0 \) \( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( 0\bigr] \) ${y}^2+{y}={x}^{3}$
51.1-a1 51.1-a \(\Q(\zeta_{9})^+\) \( 3 \cdot 17 \) $0$ $\Z/2\Z$ $1$ $1.476452004$ 0.656200891 \( -\frac{73067274470120883503414977}{188345450907} a^{2} + \frac{25375998117751504521013489}{188345450907} a + \frac{70129610583765315423118325}{62781816969} \) \( \bigl[a^{2} + a - 1\) , \( a^{2} + a - 1\) , \( 0\) , \( 417 a^{2} - 133 a - 1203\) , \( 4217 a^{2} - 237 a - 14449\bigr] \) ${y}^2+\left(a^{2}+a-1\right){x}{y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(417a^{2}-133a-1203\right){x}+4217a^{2}-237a-14449$
51.1-a2 51.1-a \(\Q(\zeta_{9})^+\) \( 3 \cdot 17 \) $0$ $\Z/4\Z$ $1$ $47.24646415$ 0.656200891 \( \frac{9123267589576143071594}{153} a^{2} + \frac{17146134462759887246327}{153} a + \frac{4854389290588414014785}{153} \) \( \bigl[a^{2} + a - 1\) , \( a^{2} + a - 1\) , \( 0\) , \( 27 a^{2} - 58 a - 43\) , \( 50 a^{2} - 62 a + 9\bigr] \) ${y}^2+\left(a^{2}+a-1\right){x}{y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(27a^{2}-58a-43\right){x}+50a^{2}-62a+9$
51.1-a3 51.1-a \(\Q(\zeta_{9})^+\) \( 3 \cdot 17 \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $11.81161603$ 0.656200891 \( -\frac{736214157382250}{60886809} a^{2} + \frac{257154008212105}{60886809} a + \frac{2126010334493071}{60886809} \) \( \bigl[a^{2} + a - 1\) , \( a^{2} + a - 1\) , \( 0\) , \( -33 a^{2} + 92 a - 33\) , \( -292 a^{2} + 546 a + 23\bigr] \) ${y}^2+\left(a^{2}+a-1\right){x}{y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(-33a^{2}+92a-33\right){x}-292a^{2}+546a+23$
51.1-a4 51.1-a \(\Q(\zeta_{9})^+\) \( 3 \cdot 17 \) $0$ $\Z/2\Z\oplus\Z/4\Z$ $1$ $94.49292831$ 0.656200891 \( \frac{21862041663547}{7803} a^{2} + \frac{41087226515522}{7803} a + \frac{3877536454165}{2601} \) \( \bigl[a^{2} + a - 1\) , \( a^{2} + a - 1\) , \( 0\) , \( -3 a^{2} + 17 a + 2\) , \( 11 a^{2} - 8 a - 6\bigr] \) ${y}^2+\left(a^{2}+a-1\right){x}{y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(-3a^{2}+17a+2\right){x}+11a^{2}-8a-6$
51.1-a5 51.1-a \(\Q(\zeta_{9})^+\) \( 3 \cdot 17 \) $0$ $\Z/8\Z$ $1$ $188.9858566$ 0.656200891 \( \frac{57227017}{153} a^{2} - \frac{97672967}{153} a - \frac{40772771}{153} \) \( \bigl[a^{2} + a - 1\) , \( a^{2} + a - 1\) , \( 0\) , \( -3 a^{2} + 17 a + 7\) , \( 17 a^{2} - 13 a - 7\bigr] \) ${y}^2+\left(a^{2}+a-1\right){x}{y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(-3a^{2}+17a+7\right){x}+17a^{2}-13a-7$
51.1-a6 51.1-a \(\Q(\zeta_{9})^+\) \( 3 \cdot 17 \) $0$ $\Z/2\Z$ $1$ $1.476452004$ 0.656200891 \( \frac{51150856611467521}{51195483} a^{2} - \frac{78367545688483633}{51195483} a - \frac{3709618270320887}{5688387} \) \( \bigl[a^{2} + a - 1\) , \( a^{2} + a - 1\) , \( 0\) , \( -963 a^{2} + 1517 a + 577\) , \( -24013 a^{2} + 36905 a + 15491\bigr] \) ${y}^2+\left(a^{2}+a-1\right){x}{y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(-963a^{2}+1517a+577\right){x}-24013a^{2}+36905a+15491$
51.2-a1 51.2-a \(\Q(\zeta_{9})^+\) \( 3 \cdot 17 \) $0$ $\Z/2\Z$ $1$ $1.476452004$ 0.656200891 \( -\frac{78367545688483633}{51195483} a^{2} + \frac{9072229692338704}{17065161} a + \frac{225650240167014325}{51195483} \) \( \bigl[a^{2} - 1\) , \( -a^{2} - a + 1\) , \( a^{2} + a - 2\) , \( 1510 a^{2} - 544 a - 4379\) , \( 37713 a^{2} - 13154 a - 108667\bigr] \) ${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{2}-a+1\right){x}^{2}+\left(1510a^{2}-544a-4379\right){x}+37713a^{2}-13154a-108667$
51.2-a2 51.2-a \(\Q(\zeta_{9})^+\) \( 3 \cdot 17 \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $11.81161603$ 0.656200891 \( \frac{257154008212105}{60886809} a^{2} + \frac{479060149170145}{60886809} a + \frac{139274003304361}{60886809} \) \( \bigl[a^{2} - 1\) , \( -a^{2} - a + 1\) , \( a^{2} + a - 2\) , \( 85 a^{2} - 49 a - 279\) , \( 599 a^{2} - 256 a - 1795\bigr] \) ${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{2}-a+1\right){x}^{2}+\left(85a^{2}-49a-279\right){x}+599a^{2}-256a-1795$
51.2-a3 51.2-a \(\Q(\zeta_{9})^+\) \( 3 \cdot 17 \) $0$ $\Z/2\Z$ $1$ $1.476452004$ 0.656200891 \( \frac{25375998117751504521013489}{188345450907} a^{2} + \frac{15897092117456459660800496}{62781816969} a + \frac{13502286575551170220498043}{188345450907} \) \( \bigl[a^{2} - 1\) , \( -a^{2} - a + 1\) , \( a^{2} + a - 2\) , \( -140 a^{2} - 274 a - 99\) , \( -139 a^{2} - 3802 a - 5503\bigr] \) ${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{2}-a+1\right){x}^{2}+\left(-140a^{2}-274a-99\right){x}-139a^{2}-3802a-5503$
51.2-a4 51.2-a \(\Q(\zeta_{9})^+\) \( 3 \cdot 17 \) $0$ $\Z/2\Z\oplus\Z/4\Z$ $1$ $94.49292831$ 0.656200891 \( \frac{41087226515522}{7803} a^{2} - \frac{20983089393023}{2601} a - \frac{26817760341455}{7803} \) \( \bigl[a^{2} - 1\) , \( -a^{2} - a + 1\) , \( a^{2} + a - 2\) , \( 10 a^{2} - 4 a - 34\) , \( -5 a^{2} + 15\bigr] \) ${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{2}-a+1\right){x}^{2}+\left(10a^{2}-4a-34\right){x}-5a^{2}+15$
51.2-a5 51.2-a \(\Q(\zeta_{9})^+\) \( 3 \cdot 17 \) $0$ $\Z/8\Z$ $1$ $188.9858566$ 0.656200891 \( -\frac{97672967}{153} a^{2} + \frac{40445950}{153} a + \frac{269027197}{153} \) \( \bigl[a^{2} - 1\) , \( -a^{2} - a + 1\) , \( a^{2} + a - 2\) , \( 10 a^{2} - 4 a - 29\) , \( -15 a^{2} + 4 a + 41\bigr] \) ${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{2}-a+1\right){x}^{2}+\left(10a^{2}-4a-29\right){x}-15a^{2}+4a+41$
51.2-a6 51.2-a \(\Q(\zeta_{9})^+\) \( 3 \cdot 17 \) $0$ $\Z/4\Z$ $1$ $47.24646415$ 0.656200891 \( \frac{17146134462759887246327}{153} a^{2} - \frac{26269402052336030317921}{153} a - \frac{11191344455779074334681}{153} \) \( \bigl[a^{2} - 1\) , \( -a^{2} - a + 1\) , \( a^{2} + a - 2\) , \( -65 a^{2} + 41 a + 131\) , \( -29 a^{2} - 60 a + 261\bigr] \) ${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{2}-a+1\right){x}^{2}+\left(-65a^{2}+41a+131\right){x}-29a^{2}-60a+261$
51.3-a1 51.3-a \(\Q(\zeta_{9})^+\) \( 3 \cdot 17 \) $0$ $\Z/2\Z$ $1$ $1.476452004$ 0.656200891 \( \frac{9072229692338704}{17065161} a^{2} + \frac{51150856611467521}{51195483} a + \frac{14481770636014835}{51195483} \) \( \bigl[a + 1\) , \( -a^{2} + a + 1\) , \( a^{2} - 2\) , \( -543 a^{2} - 968 a - 272\) , \( -13153 a^{2} - 24560 a - 6934\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+a+1\right){x}^{2}+\left(-543a^{2}-968a-272\right){x}-13153a^{2}-24560a-6934$
51.3-a2 51.3-a \(\Q(\zeta_{9})^+\) \( 3 \cdot 17 \) $0$ $\Z/4\Z$ $1$ $47.24646415$ 0.656200891 \( -\frac{26269402052336030317921}{153} a^{2} + \frac{9123267589576143071594}{153} a + \frac{75639728574412760793815}{153} \) \( \bigl[a + 1\) , \( -a^{2} + a + 1\) , \( a^{2} - 2\) , \( 42 a^{2} + 22 a - 82\) , \( -59 a^{2} + 88 a + 322\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+a+1\right){x}^{2}+\left(42a^{2}+22a-82\right){x}-59a^{2}+88a+322$
51.3-a3 51.3-a \(\Q(\zeta_{9})^+\) \( 3 \cdot 17 \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $11.81161603$ 0.656200891 \( \frac{479060149170145}{60886809} a^{2} - \frac{736214157382250}{60886809} a - \frac{304538278611719}{60886809} \) \( \bigl[a + 1\) , \( -a^{2} + a + 1\) , \( a^{2} - 2\) , \( -48 a^{2} - 38 a - 12\) , \( -255 a^{2} - 344 a - 86\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+a+1\right){x}^{2}+\left(-48a^{2}-38a-12\right){x}-255a^{2}-344a-86$
51.3-a4 51.3-a \(\Q(\zeta_{9})^+\) \( 3 \cdot 17 \) $0$ $\Z/2\Z\oplus\Z/4\Z$ $1$ $94.49292831$ 0.656200891 \( -\frac{20983089393023}{2601} a^{2} + \frac{21862041663547}{7803} a + \frac{181255229047727}{7803} \) \( \bigl[a + 1\) , \( -a^{2} + a + 1\) , \( a^{2} - 2\) , \( -3 a^{2} - 8 a - 7\) , \( a^{2} + 4 a + 4\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+a+1\right){x}^{2}+\left(-3a^{2}-8a-7\right){x}+a^{2}+4a+4$
51.3-a5 51.3-a \(\Q(\zeta_{9})^+\) \( 3 \cdot 17 \) $0$ $\Z/8\Z$ $1$ $188.9858566$ 0.656200891 \( \frac{40445950}{153} a^{2} + \frac{57227017}{153} a - \frac{7210637}{153} \) \( \bigl[a + 1\) , \( -a^{2} + a + 1\) , \( a^{2} - 2\) , \( -3 a^{2} - 8 a - 2\) , \( 5 a^{2} + 10 a + 2\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+a+1\right){x}^{2}+\left(-3a^{2}-8a-2\right){x}+5a^{2}+10a+2$
51.3-a6 51.3-a \(\Q(\zeta_{9})^+\) \( 3 \cdot 17 \) $0$ $\Z/2\Z$ $1$ $1.476452004$ 0.656200891 \( \frac{15897092117456459660800496}{62781816969} a^{2} - \frac{73067274470120883503414977}{188345450907} a - \frac{31128269893684578702277955}{188345450907} \) \( \bigl[a + 1\) , \( -a^{2} + a + 1\) , \( a^{2} - 2\) , \( -273 a^{2} + 412 a + 168\) , \( -3801 a^{2} + 3940 a + 1822\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+a+1\right){x}^{2}+\left(-273a^{2}+412a+168\right){x}-3801a^{2}+3940a+1822$
Next   Download to        

  *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.