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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 3.3.169.1 \( 2^{3} \) $0$ $\mathsf{trivial}$ $1$ $0.559664806$ 0.602715945 \( -\frac{38575685889}{16384} \) \( \bigl[a + 1\) , \( -a^{2} + a + 2\) , \( a^{2} - a - 2\) , \( 149 a^{2} - 150 a - 657\) , \( 1719 a^{2} - 1999 a - 6787\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(149a^{2}-150a-657\right){x}+1719a^{2}-1999a-6787$
8.1-a2 8.1-a 3.3.169.1 \( 2^{3} \) $0$ $\Z/7\Z$ $1$ $191.9650286$ 0.602715945 \( \frac{351}{4} \) \( \bigl[a + 1\) , \( -a^{2} + a + 2\) , \( a^{2} - a - 2\) , \( -a^{2} + 3\) , \( -a^{2} + a + 3\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(-a^{2}+3\right){x}-a^{2}+a+3$
25.1-a1 25.1-a 3.3.169.1 \( 5^{2} \) $0$ $\Z/2\Z$ $1$ $13.06442140$ 1.004955493 \( -\frac{7203238325547833397}{625} a^{2} + \frac{9176137268071346086}{625} a + \frac{26299695973412157906}{625} \) \( \bigl[a^{2} - 2\) , \( 1\) , \( a^{2} - 3\) , \( 41 a^{2} - 39 a - 177\) , \( 266 a^{2} - 322 a - 1005\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+{x}^{2}+\left(41a^{2}-39a-177\right){x}+266a^{2}-322a-1005$
25.1-a2 25.1-a 3.3.169.1 \( 5^{2} \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $52.25768563$ 1.004955493 \( -\frac{1488256999}{25} a^{2} + \frac{1889927293}{25} a + \frac{5450713881}{25} \) \( \bigl[a^{2} - 2\) , \( 1\) , \( a^{2} - 3\) , \( -4 a^{2} + 16 a - 7\) , \( -15 a^{2} + 46 a - 7\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+{x}^{2}+\left(-4a^{2}+16a-7\right){x}-15a^{2}+46a-7$
25.1-a3 25.1-a 3.3.169.1 \( 5^{2} \) $0$ $\Z/2\Z$ $1$ $52.25768563$ 1.004955493 \( 4186 a^{2} - \frac{23101}{5} a - \frac{73684}{5} \) \( \bigl[a^{2} - 2\) , \( 1\) , \( a^{2} - 3\) , \( a^{2} + a - 2\) , \( a^{2} + 2 a - 2\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+{x}^{2}+\left(a^{2}+a-2\right){x}+a^{2}+2a-2$
25.1-a4 25.1-a 3.3.169.1 \( 5^{2} \) $0$ $\Z/2\Z$ $1$ $26.12884281$ 1.004955493 \( \frac{142128188164149}{625} a^{2} - \frac{337409196609446}{625} a - \frac{103075076249858}{625} \) \( \bigl[a^{2} - 2\) , \( 1\) , \( a^{2} - 3\) , \( -129 a^{2} + 311 a + 83\) , \( -1280 a^{2} + 3050 a + 911\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+{x}^{2}+\left(-129a^{2}+311a+83\right){x}-1280a^{2}+3050a+911$
25.2-a1 25.2-a 3.3.169.1 \( 5^{2} \) $0$ $\Z/2\Z$ $1$ $26.12884281$ 1.004955493 \( -\frac{195281008445297}{625} a^{2} + \frac{49686765745289}{125} a + \frac{713871505133183}{625} \) \( \bigl[a^{2} - a - 3\) , \( -a^{2} + 2 a + 4\) , \( 0\) , \( 178 a^{2} - 227 a - 657\) , \( 2022 a^{2} - 2583 a - 7384\bigr] \) ${y}^2+\left(a^{2}-a-3\right){x}{y}={x}^{3}+\left(-a^{2}+2a+4\right){x}^{2}+\left(178a^{2}-227a-657\right){x}+2022a^{2}-2583a-7384$
25.2-a2 25.2-a 3.3.169.1 \( 5^{2} \) $0$ $\Z/2\Z$ $1$ $13.06442140$ 1.004955493 \( \frac{1972898942523512689}{625} a^{2} + \frac{3257440440500808019}{625} a + \frac{744183111721632337}{625} \) \( \bigl[a^{2} - a - 3\) , \( -a^{2} + 2 a + 4\) , \( 0\) , \( -2 a^{2} - 37 a - 47\) , \( -44 a^{2} - 177 a - 160\bigr] \) ${y}^2+\left(a^{2}-a-3\right){x}{y}={x}^{3}+\left(-a^{2}+2a+4\right){x}^{2}+\left(-2a^{2}-37a-47\right){x}-44a^{2}-177a-160$
25.2-a3 25.2-a 3.3.169.1 \( 5^{2} \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $52.25768563$ 1.004955493 \( \frac{401670294}{25} a^{2} + \frac{684916411}{25} a + \frac{182602296}{25} \) \( \bigl[a^{2} - a - 3\) , \( -a^{2} + 2 a + 4\) , \( 0\) , \( 8 a^{2} - 12 a - 32\) , \( 43 a^{2} - 60 a - 164\bigr] \) ${y}^2+\left(a^{2}-a-3\right){x}{y}={x}^{3}+\left(-a^{2}+2a+4\right){x}^{2}+\left(8a^{2}-12a-32\right){x}+43a^{2}-60a-164$
25.2-a4 25.2-a 3.3.169.1 \( 5^{2} \) $0$ $\Z/2\Z$ $1$ $52.25768563$ 1.004955493 \( -\frac{2171}{5} a^{2} - \frac{16588}{5} a - \frac{6552}{5} \) \( \bigl[a^{2} - a - 3\) , \( -a^{2} + 2 a + 4\) , \( 0\) , \( -2 a^{2} + 3 a + 8\) , \( 0\bigr] \) ${y}^2+\left(a^{2}-a-3\right){x}{y}={x}^{3}+\left(-a^{2}+2a+4\right){x}^{2}+\left(-2a^{2}+3a+8\right){x}$
25.3-a1 25.3-a 3.3.169.1 \( 5^{2} \) $0$ $\Z/2\Z$ $1$ $26.12884281$ 1.004955493 \( \frac{53152820281148}{625} a^{2} + \frac{88975367883001}{625} a + \frac{21722839234996}{625} \) \( \bigl[a\) , \( a^{2} - a - 4\) , \( a\) , \( -52 a^{2} - 78 a - 20\) , \( -511 a^{2} - 821 a - 173\bigr] \) ${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(-52a^{2}-78a-20\right){x}-511a^{2}-821a-173$
25.3-a2 25.3-a 3.3.169.1 \( 5^{2} \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $52.25768563$ 1.004955493 \( \frac{217317341}{5} a^{2} - \frac{2574843704}{25} a - \frac{785560232}{25} \) \( \bigl[a\) , \( a^{2} - a - 4\) , \( a\) , \( -7 a^{2} + 2 a + 5\) , \( -12 a^{2} - 10 a - 3\bigr] \) ${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(-7a^{2}+2a+5\right){x}-12a^{2}-10a-3$
25.3-a3 25.3-a 3.3.169.1 \( 5^{2} \) $0$ $\Z/2\Z$ $1$ $52.25768563$ 1.004955493 \( -\frac{18759}{5} a^{2} + \frac{39689}{5} a + \frac{24453}{5} \) \( \bigl[a\) , \( a^{2} - a - 4\) , \( a\) , \( -2 a^{2} + 2 a + 5\) , \( -a - 2\bigr] \) ${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(-2a^{2}+2a+5\right){x}-a-2$
25.3-a4 25.3-a 3.3.169.1 \( 5^{2} \) $0$ $\Z/2\Z$ $1$ $13.06442140$ 1.004955493 \( \frac{5230339383024320708}{625} a^{2} - \frac{2486715541714430821}{125} a - \frac{3797798826756471012}{625} \) \( \bigl[a\) , \( a^{2} - a - 4\) , \( a\) , \( -42 a^{2} + 82 a + 30\) , \( -181 a^{2} + 405 a + 123\bigr] \) ${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(-42a^{2}+82a+30\right){x}-181a^{2}+405a+123$
27.1-a1 27.1-a 3.3.169.1 \( 3^{3} \) $1$ $\mathsf{trivial}$ $1.810964640$ $1.150750920$ 0.961831951 \( -\frac{276301129}{4782969} \) \( \bigl[a^{2} - 3\) , \( -a^{2} + 2 a + 4\) , \( a^{2} - a - 3\) , \( -a^{2} - 25 a - 33\) , \( -110 a^{2} - 590 a - 605\bigr] \) ${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(-a^{2}+2a+4\right){x}^{2}+\left(-a^{2}-25a-33\right){x}-110a^{2}-590a-605$
27.1-a2 27.1-a 3.3.169.1 \( 3^{3} \) $1$ $\Z/7\Z$ $0.258709234$ $394.7075658$ 0.961831951 \( -\frac{658489}{9} \) \( \bigl[a^{2} - 3\) , \( -a^{2} + 2 a + 4\) , \( a^{2} - a - 3\) , \( -a^{2} + 2\) , \( 0\bigr] \) ${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(-a^{2}+2a+4\right){x}^{2}+\left(-a^{2}+2\right){x}$
31.1-a1 31.1-a 3.3.169.1 \( 31 \) $0$ $\Z/2\Z$ $1$ $27.63718754$ 1.062968751 \( \frac{2423516191674}{961} a^{2} - \frac{5731917800790}{961} a - \frac{1751721853153}{961} \) \( \bigl[a + 1\) , \( a^{2} - a - 3\) , \( a^{2} - 3\) , \( -4 a^{2} - 7 a - 5\) , \( 7 a^{2} + 7 a - 6\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(-4a^{2}-7a-5\right){x}+7a^{2}+7a-6$
31.1-a2 31.1-a 3.3.169.1 \( 31 \) $0$ $\Z/2\Z$ $1$ $55.27437509$ 1.062968751 \( -\frac{32365068}{31} a^{2} + \frac{42115872}{31} a + \frac{116134499}{31} \) \( \bigl[a + 1\) , \( a^{2} - a - 3\) , \( a^{2} - 3\) , \( a^{2} - 2 a - 5\) , \( a^{2} - 2 a - 5\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(a^{2}-2a-5\right){x}+a^{2}-2a-5$
31.1-b1 31.1-b 3.3.169.1 \( 31 \) $0$ $\mathsf{trivial}$ $1$ $1.439391393$ 1.107224148 \( \frac{2034273426504158658495}{819628286980801} a^{2} - \frac{2593062233763744426365}{819628286980801} a - \frac{7428286237785028686301}{819628286980801} \) \( \bigl[a + 1\) , \( -a^{2} + 2\) , \( a\) , \( 668 a^{2} - 867 a - 2461\) , \( 12546 a^{2} - 16025 a - 45866\bigr] \) ${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(668a^{2}-867a-2461\right){x}+12546a^{2}-16025a-45866$
31.1-b2 31.1-b 3.3.169.1 \( 31 \) $0$ $\Z/5\Z$ $1$ $179.9239241$ 1.107224148 \( -\frac{385258025}{961} a^{2} + \frac{915269800}{961} a + \frac{281241504}{961} \) \( \bigl[a + 1\) , \( -a^{2} + 2\) , \( a\) , \( -7 a^{2} + 8 a + 24\) , \( -4 a^{2} + 5 a + 14\bigr] \) ${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(-7a^{2}+8a+24\right){x}-4a^{2}+5a+14$
31.1-c1 31.1-c 3.3.169.1 \( 31 \) $1$ $\mathsf{trivial}$ $0.026338543$ $80.09660173$ 0.973674366 \( \frac{126638766}{961} a^{2} - \frac{161358095}{961} a - \frac{462350752}{961} \) \( \bigl[a^{2} - a - 3\) , \( -a^{2} + a + 4\) , \( 1\) , \( -3 a^{2} + 3 a + 12\) , \( -3 a^{2} + 5 a + 7\bigr] \) ${y}^2+\left(a^{2}-a-3\right){x}{y}+{y}={x}^{3}+\left(-a^{2}+a+4\right){x}^{2}+\left(-3a^{2}+3a+12\right){x}-3a^{2}+5a+7$
31.2-a1 31.2-a 3.3.169.1 \( 31 \) $0$ $\Z/2\Z$ $1$ $55.27437509$ 1.062968751 \( \frac{9750804}{31} a^{2} + \frac{12863460}{31} a - \frac{462313}{31} \) \( \bigl[a^{2} - 3\) , \( a^{2} - a - 3\) , \( 1\) , \( -a^{2} + 2 a + 2\) , \( -a^{2} - 3 a - 1\bigr] \) ${y}^2+\left(a^{2}-3\right){x}{y}+{y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(-a^{2}+2a+2\right){x}-a^{2}-3a-1$
31.2-a2 31.2-a 3.3.169.1 \( 31 \) $0$ $\Z/2\Z$ $1$ $27.63718754$ 1.062968751 \( -\frac{3308401609116}{961} a^{2} + \frac{4193287026558}{961} a + \frac{12135629940101}{961} \) \( \bigl[a^{2} - 3\) , \( a^{2} - a - 3\) , \( 1\) , \( -11 a^{2} + 27 a + 7\) , \( 29 a^{2} - 74 a - 24\bigr] \) ${y}^2+\left(a^{2}-3\right){x}{y}+{y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(-11a^{2}+27a+7\right){x}+29a^{2}-74a-24$
31.2-b1 31.2-b 3.3.169.1 \( 31 \) $0$ $\Z/5\Z$ $1$ $179.9239241$ 1.107224148 \( \frac{530011775}{961} a^{2} - \frac{674765525}{961} a - \frac{1934556121}{961} \) \( \bigl[a^{2} - 2\) , \( -a^{2} + 4\) , \( a\) , \( -a\) , \( 0\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}-a{x}$
31.2-b2 31.2-b 3.3.169.1 \( 31 \) $0$ $\mathsf{trivial}$ $1$ $1.439391393$ 1.107224148 \( -\frac{558788807259585767870}{819628286980801} a^{2} - \frac{916695811984987122755}{819628286980801} a - \frac{207888343753381175076}{819628286980801} \) \( \bigl[a^{2} - 2\) , \( -a^{2} + 4\) , \( a\) , \( -25 a^{2} + 24 a + 65\) , \( -35 a^{2} + 15 a + 45\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(-25a^{2}+24a+65\right){x}-35a^{2}+15a+45$
31.2-c1 31.2-c 3.3.169.1 \( 31 \) $1$ $\mathsf{trivial}$ $0.026338543$ $80.09660173$ 0.973674366 \( -\frac{34719329}{961} a^{2} - \frac{57200108}{961} a - \frac{12995796}{961} \) \( \bigl[a\) , \( a + 1\) , \( 1\) , \( 2 a + 3\) , \( 2 a^{2} - a - 6\bigr] \) ${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(2a+3\right){x}+2a^{2}-a-6$
31.3-a1 31.3-a 3.3.169.1 \( 31 \) $0$ $\Z/2\Z$ $1$ $27.63718754$ 1.062968751 \( \frac{884885417442}{961} a^{2} + \frac{1538630774232}{961} a + \frac{440654277869}{961} \) \( \bigl[a^{2} - a - 2\) , \( a^{2} - 4\) , \( a^{2} - a - 2\) , \( 14 a^{2} - 16 a - 55\) , \( -63 a^{2} + 81 a + 224\bigr] \) ${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(14a^{2}-16a-55\right){x}-63a^{2}+81a+224$
31.3-a2 31.3-a 3.3.169.1 \( 31 \) $0$ $\Z/2\Z$ $1$ $55.27437509$ 1.062968751 \( \frac{22614264}{31} a^{2} - \frac{54979332}{31} a - \frac{16438429}{31} \) \( \bigl[a^{2} - a - 2\) , \( a^{2} - 4\) , \( a^{2} - a - 2\) , \( -a^{2} + 4 a\) , \( -2 a^{2} + 4 a + 5\bigr] \) ${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(-a^{2}+4a\right){x}-2a^{2}+4a+5$
31.3-b1 31.3-b 3.3.169.1 \( 31 \) $0$ $\mathsf{trivial}$ $1$ $1.439391393$ 1.107224148 \( -\frac{1475484619244572890625}{819628286980801} a^{2} + \frac{3509758045748731549120}{819628286980801} a + \frac{1066714472957007302564}{819628286980801} \) \( \bigl[a^{2} - 3\) , \( a^{2} - 4\) , \( a^{2} - a - 3\) , \( 929 a^{2} - 1177 a - 3394\) , \( 21516 a^{2} - 27403 a - 78558\bigr] \) ${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(929a^{2}-1177a-3394\right){x}+21516a^{2}-27403a-78558$
31.3-b2 31.3-b 3.3.169.1 \( 31 \) $0$ $\Z/5\Z$ $1$ $179.9239241$ 1.107224148 \( -\frac{144753750}{961} a^{2} - \frac{240504275}{961} a - \frac{55013296}{961} \) \( \bigl[a^{2} - 3\) , \( a^{2} - 4\) , \( a^{2} - a - 3\) , \( 4 a^{2} - 2 a - 9\) , \( -109 a^{2} + 142 a + 402\bigr] \) ${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(4a^{2}-2a-9\right){x}-109a^{2}+142a+402$
31.3-c1 31.3-c 3.3.169.1 \( 31 \) $1$ $\mathsf{trivial}$ $0.026338543$ $80.09660173$ 0.973674366 \( -\frac{91919437}{961} a^{2} + \frac{218558203}{961} a + \frac{66685091}{961} \) \( \bigl[a^{2} - 2\) , \( a^{2} - a - 2\) , \( a^{2} - 2\) , \( 2 a^{2} + a - 2\) , \( 2 a^{2} + 2 a - 1\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(2a^{2}+a-2\right){x}+2a^{2}+2a-1$
40.1-a1 40.1-a 3.3.169.1 \( 2^{3} \cdot 5 \) $0$ $\mathsf{trivial}$ $1$ $14.96312317$ 1.151009474 \( -\frac{53626624272}{3125} a^{2} - \frac{177085593453}{6250} a - \frac{20228557469}{3125} \) \( \bigl[a + 1\) , \( -a^{2} + 2 a + 2\) , \( a^{2} - a - 2\) , \( -a^{2} - 3 a\) , \( -3 a^{2} - 6 a - 3\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{2}+2a+2\right){x}^{2}+\left(-a^{2}-3a\right){x}-3a^{2}-6a-3$
40.1-b1 40.1-b 3.3.169.1 \( 2^{3} \cdot 5 \) $0$ $\Z/3\Z$ $1$ $48.74212197$ 1.249797999 \( \frac{4300451}{250} a^{2} - \frac{3024513}{250} a - \frac{11109599}{125} \) \( \bigl[1\) , \( a^{2} - 3\) , \( a^{2} - a - 3\) , \( 12 a^{2} - 14 a - 42\) , \( -42 a^{2} + 54 a + 154\bigr] \) ${y}^2+{x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(12a^{2}-14a-42\right){x}-42a^{2}+54a+154$
40.1-b2 40.1-b 3.3.169.1 \( 2^{3} \cdot 5 \) $0$ $\mathsf{trivial}$ $1$ $1.805263776$ 1.249797999 \( -\frac{590283258329401}{40} a^{2} + \frac{1403221722682083}{40} a + \frac{428609886619833}{40} \) \( \bigl[1\) , \( a^{2} - 3\) , \( a^{2} - a - 3\) , \( -48 a^{2} + 56 a + 163\) , \( -133 a^{2} + 143 a + 446\bigr] \) ${y}^2+{x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(-48a^{2}+56a+163\right){x}-133a^{2}+143a+446$
40.2-a1 40.2-a 3.3.169.1 \( 2^{3} \cdot 5 \) $0$ $\mathsf{trivial}$ $1$ $14.96312317$ 1.151009474 \( -\frac{284338841997}{6250} a^{2} + \frac{337965466269}{3125} a + \frac{103230411712}{3125} \) \( \bigl[a^{2} - 3\) , \( -a^{2} + 2 a + 3\) , \( a^{2} - 3\) , \( -4 a^{2} + 9 a + 5\) , \( -4 a^{2} + 9 a + 4\bigr] \) ${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}+2a+3\right){x}^{2}+\left(-4a^{2}+9a+5\right){x}-4a^{2}+9a+4$
40.2-b1 40.2-b 3.3.169.1 \( 2^{3} \cdot 5 \) $0$ $\Z/3\Z$ $1$ $48.74212197$ 1.249797999 \( \frac{637969}{125} a^{2} - \frac{6852327}{250} a - \frac{11869721}{250} \) \( \bigl[1\) , \( a^{2} - 2\) , \( 0\) , \( -3 a^{2} - 4 a\) , \( 3 a^{2} + 5 a + 1\bigr] \) ${y}^2+{x}{y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(-3a^{2}-4a\right){x}+3a^{2}+5a+1$
40.2-b2 40.2-b 3.3.169.1 \( 2^{3} \cdot 5 \) $0$ $\mathsf{trivial}$ $1$ $1.805263776$ 1.249797999 \( \frac{406469232176341}{20} a^{2} - \frac{1035593670375963}{40} a - \frac{1484058408536867}{20} \) \( \bigl[1\) , \( a^{2} - 2\) , \( 0\) , \( 7 a^{2} + 36 a + 5\) , \( 51 a^{2} + 145 a + 34\bigr] \) ${y}^2+{x}{y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(7a^{2}+36a+5\right){x}+51a^{2}+145a+34$
40.3-a1 40.3-a 3.3.169.1 \( 2^{3} \cdot 5 \) $0$ $\mathsf{trivial}$ $1$ $14.96312317$ 1.151009474 \( \frac{391592090541}{6250} a^{2} - \frac{99769067817}{1250} a - \frac{1429739883649}{6250} \) \( \bigl[a^{2} - a - 2\) , \( -a^{2} + 3\) , \( a^{2} - 3\) , \( 4 a^{2} - 6 a - 15\) , \( 12 a^{2} - 16 a - 46\bigr] \) ${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(4a^{2}-6a-15\right){x}+12a^{2}-16a-46$
40.3-b1 40.3-b 3.3.169.1 \( 2^{3} \cdot 5 \) $0$ $\mathsf{trivial}$ $1$ $1.805263776$ 1.249797999 \( -\frac{222655206023281}{40} a^{2} - 9190701307653 a - \frac{41995505984563}{20} \) \( \bigl[1\) , \( a^{2} - 4\) , \( a + 1\) , \( 40 a^{2} - 87 a - 54\) , \( 107 a^{2} - 248 a - 107\bigr] \) ${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(40a^{2}-87a-54\right){x}+107a^{2}-248a-107$
40.3-b2 40.3-b 3.3.169.1 \( 2^{3} \cdot 5 \) $0$ $\Z/3\Z$ $1$ $48.74212197$ 1.249797999 \( -\frac{5576389}{250} a^{2} + \frac{987684}{25} a + \frac{3110871}{250} \) \( \bigl[1\) , \( a^{2} - 4\) , \( a + 1\) , \( -10 a^{2} + 23 a + 11\) , \( 29 a^{2} - 69 a - 23\bigr] \) ${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(-10a^{2}+23a+11\right){x}+29a^{2}-69a-23$
47.1-a1 47.1-a 3.3.169.1 \( 47 \) $1$ $\mathsf{trivial}$ $0.021269604$ $107.2059385$ 1.052412886 \( -\frac{91801}{2209} a^{2} - \frac{583557}{2209} a - \frac{629509}{2209} \) \( \bigl[1\) , \( a^{2} - 2\) , \( 1\) , \( a^{2} + a - 1\) , \( a^{2} - 2\bigr] \) ${y}^2+{x}{y}+{y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(a^{2}+a-1\right){x}+a^{2}-2$
47.2-a1 47.2-a 3.3.169.1 \( 47 \) $1$ $\mathsf{trivial}$ $0.021269604$ $107.2059385$ 1.052412886 \( -\frac{675358}{2209} a^{2} + \frac{1442517}{2209} a + \frac{445804}{2209} \) \( \bigl[1\) , \( a^{2} - 4\) , \( a\) , \( -a^{2} + a + 5\) , \( -a - 2\bigr] \) ${y}^2+{x}{y}+a{y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(-a^{2}+a+5\right){x}-a-2$
47.3-a1 47.3-a 3.3.169.1 \( 47 \) $1$ $\mathsf{trivial}$ $0.021269604$ $107.2059385$ 1.052412886 \( \frac{767159}{2209} a^{2} - \frac{858960}{2209} a - \frac{3114588}{2209} \) \( \bigl[1\) , \( a^{2} - 3\) , \( a^{2} - a - 2\) , \( -a^{2} + 3 a + 4\) , \( a - 1\bigr] \) ${y}^2+{x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(-a^{2}+3a+4\right){x}+a-1$
64.1-a1 64.1-a 3.3.169.1 \( 2^{6} \) $0$ $\Z/2\Z$ $1$ $82.94864304$ 1.595166212 \( 115659804096 a^{2} + 190965142512 a + 43627210848 \) \( \bigl[0\) , \( 0\) , \( 0\) , \( -155 a^{2} - 255 a - 59\) , \( 2546 a^{2} + 4202 a + 960\bigr] \) ${y}^2={x}^{3}+\left(-155a^{2}-255a-59\right){x}+2546a^{2}+4202a+960$
64.1-a2 64.1-a 3.3.169.1 \( 2^{6} \) $0$ $\Z/2\Z$ $1$ $82.94864304$ 1.595166212 \( -422284750704 a^{2} + 537944554800 a + 1541801071152 \) \( \bigl[0\) , \( 0\) , \( 0\) , \( -5 a^{2} - 15 a - 14\) , \( 54 a^{2} + 100 a + 36\bigr] \) ${y}^2={x}^{3}+\left(-5a^{2}-15a-14\right){x}+54a^{2}+100a+36$
64.1-a3 64.1-a 3.3.169.1 \( 2^{6} \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $331.7945721$ 1.595166212 \( 1168128 \) \( \bigl[0\) , \( 0\) , \( 0\) , \( -10 a^{2} - 15 a - 4\) , \( 40 a^{2} + 65 a + 15\bigr] \) ${y}^2={x}^{3}+\left(-10a^{2}-15a-4\right){x}+40a^{2}+65a+15$
64.1-a4 64.1-a 3.3.169.1 \( 2^{6} \) $0$ $\Z/2\Z$ $1$ $82.94864304$ 1.595166212 \( 306624946608 a^{2} - 728909697312 a - 222643270080 \) \( \bigl[0\) , \( 0\) , \( 0\) , \( -20 a^{2} + 1\) , \( 40 a^{2} - 12 a - 6\bigr] \) ${y}^2={x}^{3}+\left(-20a^{2}+1\right){x}+40a^{2}-12a-6$
64.1-b1 64.1-b 3.3.169.1 \( 2^{6} \) $0$ $\mathsf{trivial}$ $1$ $0.447583986$ 0.309865836 \( -368484688 \) \( \bigl[0\) , \( -a^{2} + 2 a + 4\) , \( 0\) , \( -a^{2} - 200 a - 278\) , \( -329 a^{2} - 1965 a - 2085\bigr] \) ${y}^2={x}^{3}+\left(-a^{2}+2a+4\right){x}^{2}+\left(-a^{2}-200a-278\right){x}-329a^{2}-1965a-2085$
64.1-b2 64.1-b 3.3.169.1 \( 2^{6} \) $0$ $\Z/3\Z$ $1$ $12.08476763$ 0.309865836 \( -208 \) \( \bigl[0\) , \( -a^{2} + 2 a + 4\) , \( 0\) , \( -a^{2} + 2\) , \( -a^{2} - 5 a - 5\bigr] \) ${y}^2={x}^{3}+\left(-a^{2}+2a+4\right){x}^{2}+\left(-a^{2}+2\right){x}-a^{2}-5a-5$
65.1-a1 65.1-a 3.3.169.1 \( 5 \cdot 13 \) $1$ $\Z/2\Z$ $0.194885731$ $115.5814743$ 1.299529626 \( \frac{4431229}{65} a^{2} - \frac{828994}{13} a - \frac{14115071}{65} \) \( \bigl[a + 1\) , \( 1\) , \( a^{2} - 3\) , \( -2 a^{2} - 2 a + 1\) , \( -4 a^{2} - 8 a - 4\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+{x}^{2}+\left(-2a^{2}-2a+1\right){x}-4a^{2}-8a-4$
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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.