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Label Class Base field Conductor norm Rank Torsion CM Regulator Period Leading coeff j-invariant Weierstrass coefficients Weierstrass equation
1.1-a1 1.1-a \(\Q(\zeta_{13})^+\) \( 1 \) $0$ $\mathsf{trivial}$ $1$ \( -3387888351672962316333 a^{4} + 3387888351672962316333 a^{3} + 13551553406691849265332 a^{2} - 6775776703345924632666 a - 14577323462934449612494 \) \( \bigl[a^{5} - 4 a^{3} + a^{2} + 2 a - 2\) , \( -a^{5} + 6 a^{3} - 8 a + 1\) , \( a^{3} + a^{2} - 2 a - 1\) , \( -6649 a^{5} + 3526 a^{4} + 36471 a^{3} - 7581 a^{2} - 48724 a - 11069\) , \( -549202 a^{5} + 227195 a^{4} + 2939749 a^{3} - 400010 a^{2} - 3734843 a - 839159\bigr] \) ${y}^2+\left(a^{5}-4a^{3}+a^{2}+2a-2\right){x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(-a^{5}+6a^{3}-8a+1\right){x}^{2}+\left(-6649a^{5}+3526a^{4}+36471a^{3}-7581a^{2}-48724a-11069\right){x}-549202a^{5}+227195a^{4}+2939749a^{3}-400010a^{2}-3734843a-839159$
1.1-a2 1.1-a \(\Q(\zeta_{13})^+\) \( 1 \) $0$ $\Z/19\Z$ $1$ \( -17787 a^{4} + 17787 a^{3} + 71148 a^{2} - 35574 a - 75349 \) \( \bigl[a^{4} + a^{3} - 3 a^{2} - 2 a + 1\) , \( a^{5} - a^{4} - 5 a^{3} + 3 a^{2} + 5 a\) , \( a^{4} + a^{3} - 4 a^{2} - 3 a + 3\) , \( 5 a^{5} - 18 a^{3} + 2 a^{2} + 12 a\) , \( 8 a^{5} + 4 a^{4} - 27 a^{3} - 7 a^{2} + 16 a + 1\bigr] \) ${y}^2+\left(a^{4}+a^{3}-3a^{2}-2a+1\right){x}{y}+\left(a^{4}+a^{3}-4a^{2}-3a+3\right){y}={x}^{3}+\left(a^{5}-a^{4}-5a^{3}+3a^{2}+5a\right){x}^{2}+\left(5a^{5}-18a^{3}+2a^{2}+12a\right){x}+8a^{5}+4a^{4}-27a^{3}-7a^{2}+16a+1$
1.1-a3 1.1-a \(\Q(\zeta_{13})^+\) \( 1 \) $0$ $\Z/19\Z$ $1$ \( 17787 a^{4} - 17787 a^{3} - 71148 a^{2} + 35574 a + 13586 \) \( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( -a^{5} + 6 a^{3} + a^{2} - 8 a - 1\) , \( a^{5} + a^{4} - 5 a^{3} - 4 a^{2} + 5 a + 3\) , \( 2 a^{5} - 9 a^{3} + 9 a + 3\) , \( a^{4} + a^{3} - 2 a^{2} - 3 a - 1\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-1\right){x}{y}+\left(a^{5}+a^{4}-5a^{3}-4a^{2}+5a+3\right){y}={x}^{3}+\left(-a^{5}+6a^{3}+a^{2}-8a-1\right){x}^{2}+\left(2a^{5}-9a^{3}+9a+3\right){x}+a^{4}+a^{3}-2a^{2}-3a-1$
1.1-a4 1.1-a \(\Q(\zeta_{13})^+\) \( 1 \) $0$ $\mathsf{trivial}$ $1$ \( 3387888351672962316333 a^{4} - 3387888351672962316333 a^{3} - 13551553406691849265332 a^{2} + 6775776703345924632666 a + 2362118295430361969171 \) \( \bigl[a^{5} + a^{4} - 5 a^{3} - 4 a^{2} + 6 a + 2\) , \( -a^{4} - a^{3} + 5 a^{2} + 4 a - 5\) , \( a^{5} + a^{4} - 5 a^{3} - 3 a^{2} + 6 a + 1\) , \( 9022 a^{5} + 849 a^{4} - 39313 a^{3} - 5670 a^{2} + 26993 a - 4629\) , \( 611845 a^{5} + 158368 a^{4} - 2641015 a^{3} - 821368 a^{2} + 1693284 a - 462\bigr] \) ${y}^2+\left(a^{5}+a^{4}-5a^{3}-4a^{2}+6a+2\right){x}{y}+\left(a^{5}+a^{4}-5a^{3}-3a^{2}+6a+1\right){y}={x}^{3}+\left(-a^{4}-a^{3}+5a^{2}+4a-5\right){x}^{2}+\left(9022a^{5}+849a^{4}-39313a^{3}-5670a^{2}+26993a-4629\right){x}+611845a^{5}+158368a^{4}-2641015a^{3}-821368a^{2}+1693284a-462$
27.1-a1 27.1-a \(\Q(\zeta_{13})^+\) \( 3^{3} \) $0$ $\mathsf{trivial}$ $1$ \( -\frac{219992945997349946}{2187} a^{4} + \frac{219992945997349946}{2187} a^{3} + \frac{879971783989399784}{2187} a^{2} - \frac{439985891994699892}{2187} a - \frac{315526762837596247}{729} \) \( \bigl[a^{4} + a^{3} - 3 a^{2} - 3 a\) , \( -a^{5} + a^{4} + 4 a^{3} - 5 a^{2} - 3 a + 5\) , \( a^{5} + a^{4} - 5 a^{3} - 3 a^{2} + 5 a\) , \( -10 a^{5} + 135 a^{4} + 187 a^{3} - 465 a^{2} - 620 a - 115\) , \( 1078 a^{5} + 2070 a^{4} - 2830 a^{3} - 7811 a^{2} - 4432 a - 668\bigr] \) ${y}^2+\left(a^{4}+a^{3}-3a^{2}-3a\right){x}{y}+\left(a^{5}+a^{4}-5a^{3}-3a^{2}+5a\right){y}={x}^{3}+\left(-a^{5}+a^{4}+4a^{3}-5a^{2}-3a+5\right){x}^{2}+\left(-10a^{5}+135a^{4}+187a^{3}-465a^{2}-620a-115\right){x}+1078a^{5}+2070a^{4}-2830a^{3}-7811a^{2}-4432a-668$
27.1-a2 27.1-a \(\Q(\zeta_{13})^+\) \( 3^{3} \) $0$ $\Z/7\Z$ $1$ \( \frac{7918}{3} a^{4} - \frac{7918}{3} a^{3} - \frac{31672}{3} a^{2} + \frac{15836}{3} a + 1901 \) \( \bigl[a^{5} + a^{4} - 5 a^{3} - 4 a^{2} + 6 a + 2\) , \( a^{5} - a^{4} - 5 a^{3} + 3 a^{2} + 4 a + 1\) , \( a^{5} + a^{4} - 5 a^{3} - 4 a^{2} + 5 a + 3\) , \( 3 a^{5} - 2 a^{4} - 13 a^{3} + 5 a^{2} + 7 a + 3\) , \( 3 a^{5} - 4 a^{4} - 13 a^{3} + 14 a^{2} + 10 a - 5\bigr] \) ${y}^2+\left(a^{5}+a^{4}-5a^{3}-4a^{2}+6a+2\right){x}{y}+\left(a^{5}+a^{4}-5a^{3}-4a^{2}+5a+3\right){y}={x}^{3}+\left(a^{5}-a^{4}-5a^{3}+3a^{2}+4a+1\right){x}^{2}+\left(3a^{5}-2a^{4}-13a^{3}+5a^{2}+7a+3\right){x}+3a^{5}-4a^{4}-13a^{3}+14a^{2}+10a-5$
27.1-b1 27.1-b \(\Q(\zeta_{13})^+\) \( 3^{3} \) $0$ $\mathsf{trivial}$ $1$ \( \frac{9799933}{19683} a^{4} - \frac{9799933}{19683} a^{3} - \frac{39199732}{19683} a^{2} + \frac{19599866}{19683} a + \frac{13454051}{6561} \) \( \bigl[a^{5} - 5 a^{3} + a^{2} + 5 a - 1\) , \( a^{5} - 4 a^{3} - a^{2} + a + 3\) , \( a^{4} - 4 a^{2} + 3\) , \( -a^{5} + a^{4} + 2 a^{3} - 3 a^{2} - a + 8\) , \( a^{4} - 4 a^{3} - 2 a^{2} + 6 a + 2\bigr] \) ${y}^2+\left(a^{5}-5a^{3}+a^{2}+5a-1\right){x}{y}+\left(a^{4}-4a^{2}+3\right){y}={x}^{3}+\left(a^{5}-4a^{3}-a^{2}+a+3\right){x}^{2}+\left(-a^{5}+a^{4}+2a^{3}-3a^{2}-a+8\right){x}+a^{4}-4a^{3}-2a^{2}+6a+2$
27.1-b2 27.1-b \(\Q(\zeta_{13})^+\) \( 3^{3} \) $0$ $\mathsf{trivial}$ $1$ \( \frac{1173836521}{27} a^{4} - \frac{1173836521}{27} a^{3} - \frac{4695346084}{27} a^{2} + \frac{2347673042}{27} a + \frac{272732408}{9} \) \( \bigl[a^{5} - 4 a^{3} + a^{2} + 3 a - 1\) , \( a^{5} - a^{4} - 4 a^{3} + 5 a^{2} + 2 a - 5\) , \( a^{5} + a^{4} - 4 a^{3} - 4 a^{2} + 3 a + 3\) , \( 91 a^{5} + 53 a^{4} - 378 a^{3} - 238 a^{2} + 196 a + 60\) , \( 788 a^{5} + 381 a^{4} - 3365 a^{3} - 1823 a^{2} + 1998 a + 545\bigr] \) ${y}^2+\left(a^{5}-4a^{3}+a^{2}+3a-1\right){x}{y}+\left(a^{5}+a^{4}-4a^{3}-4a^{2}+3a+3\right){y}={x}^{3}+\left(a^{5}-a^{4}-4a^{3}+5a^{2}+2a-5\right){x}^{2}+\left(91a^{5}+53a^{4}-378a^{3}-238a^{2}+196a+60\right){x}+788a^{5}+381a^{4}-3365a^{3}-1823a^{2}+1998a+545$
27.2-a1 27.2-a \(\Q(\zeta_{13})^+\) \( 3^{3} \) $0$ $\Z/7\Z$ $1$ \( -\frac{7918}{3} a^{4} + \frac{7918}{3} a^{3} + \frac{31672}{3} a^{2} - \frac{15836}{3} a - \frac{33887}{3} \) \( \bigl[a^{5} - 4 a^{3} + a^{2} + 2 a - 2\) , \( a^{5} - a^{4} - 4 a^{3} + 4 a^{2} + a - 1\) , \( a^{5} - 4 a^{3} + 2 a + 1\) , \( -a^{4} + a^{3} + 5 a^{2} - 6 a - 1\) , \( 2 a^{5} - 9 a^{3} + 5 a + 1\bigr] \) ${y}^2+\left(a^{5}-4a^{3}+a^{2}+2a-2\right){x}{y}+\left(a^{5}-4a^{3}+2a+1\right){y}={x}^{3}+\left(a^{5}-a^{4}-4a^{3}+4a^{2}+a-1\right){x}^{2}+\left(-a^{4}+a^{3}+5a^{2}-6a-1\right){x}+2a^{5}-9a^{3}+5a+1$
27.2-a2 27.2-a \(\Q(\zeta_{13})^+\) \( 3^{3} \) $0$ $\mathsf{trivial}$ $1$ \( \frac{219992945997349946}{2187} a^{4} - \frac{219992945997349946}{2187} a^{3} - \frac{879971783989399784}{2187} a^{2} + \frac{439985891994699892}{2187} a + \frac{153384441473960989}{2187} \) \( \bigl[a^{5} - 5 a^{3} + a^{2} + 6 a - 2\) , \( -a^{5} + 4 a^{3} + a^{2} - 2 a - 1\) , \( a^{4} - 3 a^{2} + a\) , \( 294 a^{5} - 147 a^{4} - 1315 a^{3} + 565 a^{2} + 1084 a - 640\) , \( 4701 a^{5} - 3219 a^{4} - 21365 a^{3} + 12804 a^{2} + 18756 a - 12305\bigr] \) ${y}^2+\left(a^{5}-5a^{3}+a^{2}+6a-2\right){x}{y}+\left(a^{4}-3a^{2}+a\right){y}={x}^{3}+\left(-a^{5}+4a^{3}+a^{2}-2a-1\right){x}^{2}+\left(294a^{5}-147a^{4}-1315a^{3}+565a^{2}+1084a-640\right){x}+4701a^{5}-3219a^{4}-21365a^{3}+12804a^{2}+18756a-12305$
27.2-b1 27.2-b \(\Q(\zeta_{13})^+\) \( 3^{3} \) $0$ $\mathsf{trivial}$ $1$ \( -\frac{1173836521}{27} a^{4} + \frac{1173836521}{27} a^{3} + \frac{4695346084}{27} a^{2} - \frac{2347673042}{27} a - \frac{5050985381}{27} \) \( \bigl[a^{4} - 4 a^{2} + 3\) , \( -a^{5} + a^{4} + 6 a^{3} - 3 a^{2} - 7 a + 1\) , \( a^{5} + a^{4} - 4 a^{3} - 3 a^{2} + 2 a\) , \( 105 a^{5} + 58 a^{4} - 441 a^{3} - 268 a^{2} + 237 a + 70\) , \( -1048 a^{5} - 518 a^{4} + 4466 a^{3} + 2478 a^{2} - 2587 a - 709\bigr] \) ${y}^2+\left(a^{4}-4a^{2}+3\right){x}{y}+\left(a^{5}+a^{4}-4a^{3}-3a^{2}+2a\right){y}={x}^{3}+\left(-a^{5}+a^{4}+6a^{3}-3a^{2}-7a+1\right){x}^{2}+\left(105a^{5}+58a^{4}-441a^{3}-268a^{2}+237a+70\right){x}-1048a^{5}-518a^{4}+4466a^{3}+2478a^{2}-2587a-709$
27.2-b2 27.2-b \(\Q(\zeta_{13})^+\) \( 3^{3} \) $0$ $\mathsf{trivial}$ $1$ \( -\frac{9799933}{19683} a^{4} + \frac{9799933}{19683} a^{3} + \frac{39199732}{19683} a^{2} - \frac{19599866}{19683} a - \frac{8637512}{19683} \) \( \bigl[a^{4} + a^{3} - 4 a^{2} - 3 a + 3\) , \( -a^{5} + 5 a^{3} - a^{2} - 6 a + 1\) , \( a^{5} - 5 a^{3} + a^{2} + 5 a - 1\) , \( -a^{5} + a^{4} + 6 a^{3} - 6 a^{2} - 9 a + 9\) , \( a^{5} - 3 a^{4} - 4 a^{3} + 10 a^{2} + 3 a - 6\bigr] \) ${y}^2+\left(a^{4}+a^{3}-4a^{2}-3a+3\right){x}{y}+\left(a^{5}-5a^{3}+a^{2}+5a-1\right){y}={x}^{3}+\left(-a^{5}+5a^{3}-a^{2}-6a+1\right){x}^{2}+\left(-a^{5}+a^{4}+6a^{3}-6a^{2}-9a+9\right){x}+a^{5}-3a^{4}-4a^{3}+10a^{2}+3a-6$
53.1-a1 53.1-a \(\Q(\zeta_{13})^+\) \( 53 \) $1$ $\mathsf{trivial}$ $0.003837573$ \( \frac{1343121899}{53} a^{5} - \frac{1680516771}{53} a^{4} - \frac{6274189252}{53} a^{3} + \frac{6889341303}{53} a^{2} + \frac{6343404041}{53} a - \frac{5544409259}{53} \) \( \bigl[a^{5} + a^{4} - 5 a^{3} - 4 a^{2} + 5 a + 3\) , \( -a^{3} - a^{2} + 3 a + 3\) , \( a^{3} + a^{2} - 2 a - 2\) , \( 3 a^{5} - 3 a^{4} - 17 a^{3} + 7 a^{2} + 23 a + 5\) , \( a^{5} - a^{4} - 7 a^{3} + 2 a^{2} + 11 a + 2\bigr] \) ${y}^2+\left(a^{5}+a^{4}-5a^{3}-4a^{2}+5a+3\right){x}{y}+\left(a^{3}+a^{2}-2a-2\right){y}={x}^{3}+\left(-a^{3}-a^{2}+3a+3\right){x}^{2}+\left(3a^{5}-3a^{4}-17a^{3}+7a^{2}+23a+5\right){x}+a^{5}-a^{4}-7a^{3}+2a^{2}+11a+2$
53.1-a2 53.1-a \(\Q(\zeta_{13})^+\) \( 53 \) $1$ $\mathsf{trivial}$ $0.001279191$ \( \frac{15669618624}{148877} a^{5} + \frac{14755246679}{148877} a^{4} - \frac{49850712165}{148877} a^{3} - \frac{33922667128}{148877} a^{2} + \frac{28694128602}{148877} a + \frac{8142934591}{148877} \) \( \bigl[a^{2} + a - 1\) , \( a^{5} - a^{4} - 6 a^{3} + 3 a^{2} + 9 a + 1\) , \( a^{5} + a^{4} - 5 a^{3} - 4 a^{2} + 5 a + 3\) , \( a^{5} - 2 a^{4} - 8 a^{3} + 6 a^{2} + 17 a + 4\) , \( 2 a^{5} - a^{4} - 11 a^{3} + 2 a^{2} + 15 a + 4\bigr] \) ${y}^2+\left(a^{2}+a-1\right){x}{y}+\left(a^{5}+a^{4}-5a^{3}-4a^{2}+5a+3\right){y}={x}^{3}+\left(a^{5}-a^{4}-6a^{3}+3a^{2}+9a+1\right){x}^{2}+\left(a^{5}-2a^{4}-8a^{3}+6a^{2}+17a+4\right){x}+2a^{5}-a^{4}-11a^{3}+2a^{2}+15a+4$
53.2-a1 53.2-a \(\Q(\zeta_{13})^+\) \( 53 \) $1$ $\mathsf{trivial}$ $0.003837573$ \( -\frac{510635032}{53} a^{5} - \frac{391066624}{53} a^{4} + \frac{1601119885}{53} a^{3} + \frac{835805000}{53} a^{2} - \frac{816338829}{53} a - \frac{221982260}{53} \) \( \bigl[a^{4} - 3 a^{2} + 1\) , \( a^{5} - 6 a^{3} + 8 a - 1\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -a^{4} - a^{3} + a^{2} + 2 a + 3\) , \( -a^{4} + 3 a^{2} - 1\bigr] \) ${y}^2+\left(a^{4}-3a^{2}+1\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{5}-6a^{3}+8a-1\right){x}^{2}+\left(-a^{4}-a^{3}+a^{2}+2a+3\right){x}-a^{4}+3a^{2}-1$
53.2-a2 53.2-a \(\Q(\zeta_{13})^+\) \( 53 \) $1$ $\mathsf{trivial}$ $0.001279191$ \( -\frac{7340797392}{148877} a^{5} + \frac{20168559723}{148877} a^{4} + \frac{865808613}{148877} a^{3} - \frac{30080813866}{148877} a^{2} + \frac{9874050146}{148877} a + \frac{3817528150}{148877} \) \( \bigl[a^{5} + a^{4} - 4 a^{3} - 3 a^{2} + 2 a\) , \( a^{5} + a^{4} - 4 a^{3} - 4 a^{2} + 2 a + 1\) , \( a^{5} + a^{4} - 4 a^{3} - 3 a^{2} + 2 a + 1\) , \( 2 a^{5} + a^{4} - 7 a^{3} - 4 a^{2} + 3 a + 1\) , \( 2 a^{5} + a^{4} - 7 a^{3} - 3 a^{2} + 4 a + 1\bigr] \) ${y}^2+\left(a^{5}+a^{4}-4a^{3}-3a^{2}+2a\right){x}{y}+\left(a^{5}+a^{4}-4a^{3}-3a^{2}+2a+1\right){y}={x}^{3}+\left(a^{5}+a^{4}-4a^{3}-4a^{2}+2a+1\right){x}^{2}+\left(2a^{5}+a^{4}-7a^{3}-4a^{2}+3a+1\right){x}+2a^{5}+a^{4}-7a^{3}-3a^{2}+4a+1$
53.3-a1 53.3-a \(\Q(\zeta_{13})^+\) \( 53 \) $1$ $\mathsf{trivial}$ $0.003837573$ \( -\frac{1510396118}{53} a^{5} + \frac{391066624}{53} a^{4} + \frac{7889375462}{53} a^{3} - \frac{221144597}{53} a^{2} - \frac{9465866862}{53} a - \frac{2409815930}{53} \) \( \bigl[a^{5} + a^{4} - 5 a^{3} - 3 a^{2} + 6 a\) , \( a^{3} - 3 a + 1\) , \( a^{4} - 3 a^{2}\) , \( 3 a^{5} - 8 a^{3} + 2 a^{2} - a - 1\) , \( 5 a^{5} + a^{4} - 17 a^{3} - a^{2} + 9 a + 2\bigr] \) ${y}^2+\left(a^{5}+a^{4}-5a^{3}-3a^{2}+6a\right){x}{y}+\left(a^{4}-3a^{2}\right){y}={x}^{3}+\left(a^{3}-3a+1\right){x}^{2}+\left(3a^{5}-8a^{3}+2a^{2}-a-1\right){x}+5a^{5}+a^{4}-17a^{3}-a^{2}+9a+2$
53.3-a2 53.3-a \(\Q(\zeta_{13})^+\) \( 53 \) $1$ $\mathsf{trivial}$ $0.001279191$ \( -\frac{40767938212}{148877} a^{5} - \frac{20168559723}{148877} a^{4} + \frac{173414825757}{148877} a^{3} + \frac{96343857516}{148877} a^{2} - \frac{99737332820}{148877} a - \frac{27177657661}{148877} \) \( \bigl[a^{5} + a^{4} - 5 a^{3} - 4 a^{2} + 5 a + 3\) , \( -a^{5} - a^{4} + 5 a^{3} + 5 a^{2} - 4 a - 5\) , \( a\) , \( -4 a^{5} + 17 a^{3} + 2 a^{2} - 12 a + 1\) , \( -a^{5} - a^{4} + 4 a^{3} + 4 a^{2} - a - 1\bigr] \) ${y}^2+\left(a^{5}+a^{4}-5a^{3}-4a^{2}+5a+3\right){x}{y}+a{y}={x}^{3}+\left(-a^{5}-a^{4}+5a^{3}+5a^{2}-4a-5\right){x}^{2}+\left(-4a^{5}+17a^{3}+2a^{2}-12a+1\right){x}-a^{5}-a^{4}+4a^{3}+4a^{2}-a-1$
53.4-a1 53.4-a \(\Q(\zeta_{13})^+\) \( 53 \) $1$ $\mathsf{trivial}$ $0.001279191$ \( -\frac{43252627634}{148877} a^{5} + \frac{53595700543}{148877} a^{4} + \frac{203435375839}{148877} a^{3} - \frac{221723599564}{148877} a^{2} - \frac{206277232132}{148877} a + \frac{179199457234}{148877} \) \( \bigl[a^{4} - 3 a^{2} + 1\) , \( a^{5} - a^{4} - 4 a^{3} + 4 a^{2} + 2 a - 1\) , \( a^{2} + a - 2\) , \( a^{5} - a^{4} - 3 a^{3} + 4 a^{2} - a\) , \( a^{5} - a^{4} - 3 a^{3} + 4 a^{2} - 1\bigr] \) ${y}^2+\left(a^{4}-3a^{2}+1\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{5}-a^{4}-4a^{3}+4a^{2}+2a-1\right){x}^{2}+\left(a^{5}-a^{4}-3a^{3}+4a^{2}-a\right){x}+a^{5}-a^{4}-3a^{3}+4a^{2}-1$
53.4-a2 53.4-a \(\Q(\zeta_{13})^+\) \( 53 \) $1$ $\mathsf{trivial}$ $0.003837573$ \( \frac{1239096528}{53} a^{5} + \frac{608694462}{53} a^{4} - \frac{5293780984}{53} a^{3} - \frac{2945412880}{53} a^{2} + \frac{3048957429}{53} a + \frac{831251952}{53} \) \( \bigl[a^{5} - 4 a^{3} + 2 a + 1\) , \( a^{5} + a^{4} - 4 a^{3} - 3 a^{2} + 3 a\) , \( a^{2} - 1\) , \( 2 a^{5} + 2 a^{4} - 7 a^{3} - 5 a^{2} + 5 a + 1\) , \( a^{5} + a^{4} - 2 a^{3} - a^{2} + a\bigr] \) ${y}^2+\left(a^{5}-4a^{3}+2a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{5}+a^{4}-4a^{3}-3a^{2}+3a\right){x}^{2}+\left(2a^{5}+2a^{4}-7a^{3}-5a^{2}+5a+1\right){x}+a^{5}+a^{4}-2a^{3}-a^{2}+a$
53.5-a1 53.5-a \(\Q(\zeta_{13})^+\) \( 53 \) $1$ $\mathsf{trivial}$ $0.003837573$ \( \frac{167274219}{53} a^{5} - \frac{608694462}{53} a^{4} + \frac{282958399}{53} a^{3} + \frac{924381730}{53} a^{2} - \frac{673826397}{53} a - \frac{212698186}{53} \) \( \bigl[a^{2} + a - 1\) , \( -a^{5} - a^{4} + 4 a^{3} + 4 a^{2} - a - 3\) , \( a^{5} - 4 a^{3} + a^{2} + 3 a - 1\) , \( -5 a^{5} - 3 a^{4} + 21 a^{3} + 12 a^{2} - 14 a - 4\) , \( -4 a^{5} - 2 a^{4} + 15 a^{3} + 9 a^{2} - 7 a - 2\bigr] \) ${y}^2+\left(a^{2}+a-1\right){x}{y}+\left(a^{5}-4a^{3}+a^{2}+3a-1\right){y}={x}^{3}+\left(-a^{5}-a^{4}+4a^{3}+4a^{2}-a-3\right){x}^{2}+\left(-5a^{5}-3a^{4}+21a^{3}+12a^{2}-14a-4\right){x}-4a^{5}-2a^{4}+15a^{3}+9a^{2}-7a-2$
53.5-a2 53.5-a \(\Q(\zeta_{13})^+\) \( 53 \) $1$ $\mathsf{trivial}$ $0.001279191$ \( \frac{25098319588}{148877} a^{5} - \frac{53595700543}{148877} a^{4} - \frac{64555100005}{148877} a^{3} + \frac{173614863960}{148877} a^{2} - \frac{46974822956}{148877} a - \frac{21924763841}{148877} \) \( \bigl[a^{5} + a^{4} - 5 a^{3} - 3 a^{2} + 6 a\) , \( -a^{5} + a^{4} + 5 a^{3} - 5 a^{2} - 6 a + 5\) , \( a^{2} + a - 1\) , \( -3 a^{5} + 2 a^{4} + 15 a^{3} - 10 a^{2} - 17 a + 12\) , \( -2 a^{5} + 3 a^{4} + 10 a^{3} - 13 a^{2} - 12 a + 12\bigr] \) ${y}^2+\left(a^{5}+a^{4}-5a^{3}-3a^{2}+6a\right){x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+\left(-a^{5}+a^{4}+5a^{3}-5a^{2}-6a+5\right){x}^{2}+\left(-3a^{5}+2a^{4}+15a^{3}-10a^{2}-17a+12\right){x}-2a^{5}+3a^{4}+10a^{3}-13a^{2}-12a+12$
53.6-a1 53.6-a \(\Q(\zeta_{13})^+\) \( 53 \) $1$ $\mathsf{trivial}$ $0.001279191$ \( \frac{50593425026}{148877} a^{5} - \frac{14755246679}{148877} a^{4} - \frac{263310198039}{148877} a^{3} + \frac{15768359082}{148877} a^{2} + \frac{314421209160}{148877} a + \frac{71068595640}{148877} \) \( \bigl[a^{5} - 4 a^{3} + 2 a + 1\) , \( a^{5} + a^{4} - 6 a^{3} - 3 a^{2} + 8 a - 1\) , \( a^{4} - 3 a^{2} + 1\) , \( -a^{4} + a^{3} + 3 a^{2} - 3 a + 1\) , \( 0\bigr] \) ${y}^2+\left(a^{5}-4a^{3}+2a+1\right){x}{y}+\left(a^{4}-3a^{2}+1\right){y}={x}^{3}+\left(a^{5}+a^{4}-6a^{3}-3a^{2}+8a-1\right){x}^{2}+\left(-a^{4}+a^{3}+3a^{2}-3a+1\right){x}$
53.6-a2 53.6-a \(\Q(\zeta_{13})^+\) \( 53 \) $1$ $\mathsf{trivial}$ $0.003837573$ \( -\frac{728461496}{53} a^{5} + \frac{1680516771}{53} a^{4} + \frac{1794516490}{53} a^{3} - \frac{5482970556}{53} a^{2} + \frac{1563670618}{53} a + \frac{714529978}{53} \) \( \bigl[a^{5} + a^{4} - 4 a^{3} - 3 a^{2} + 2 a\) , \( a^{5} - 4 a^{3} + a\) , \( a^{5} - 4 a^{3} + a^{2} + 2 a - 2\) , \( a^{5} + 2 a^{4} - 4 a^{3} - 7 a^{2} + a + 2\) , \( a^{5} - 4 a^{3} + a^{2} + 3 a - 2\bigr] \) ${y}^2+\left(a^{5}+a^{4}-4a^{3}-3a^{2}+2a\right){x}{y}+\left(a^{5}-4a^{3}+a^{2}+2a-2\right){y}={x}^{3}+\left(a^{5}-4a^{3}+a\right){x}^{2}+\left(a^{5}+2a^{4}-4a^{3}-7a^{2}+a+2\right){x}+a^{5}-4a^{3}+a^{2}+3a-2$
64.1-a1 64.1-a \(\Q(\zeta_{13})^+\) \( 2^{6} \) $0$ $\mathsf{trivial}$ $1$ \( -\frac{1250637664527933}{32} a^{4} + \frac{1250637664527933}{32} a^{3} + \frac{1250637664527933}{8} a^{2} - \frac{1250637664527933}{16} a - \frac{2690606637259811}{16} \) \( \bigl[1\) , \( 1\) , \( a^{4} + a^{3} - 4 a^{2} - 2 a + 3\) , \( -29 a^{4} + 28 a^{3} + 116 a^{2} - 56 a - 85\) , \( -53 a^{4} + 51 a^{3} + 211 a^{2} - 102 a - 263\bigr] \) ${y}^2+{x}{y}+\left(a^{4}+a^{3}-4a^{2}-2a+3\right){y}={x}^{3}+{x}^{2}+\left(-29a^{4}+28a^{3}+116a^{2}-56a-85\right){x}-53a^{4}+51a^{3}+211a^{2}-102a-263$
64.1-a2 64.1-a \(\Q(\zeta_{13})^+\) \( 2^{6} \) $0$ $\Z/5\Z$ $1$ \( -\frac{461373}{2} a^{4} + \frac{461373}{2} a^{3} + 922746 a^{2} - 461373 a - 992771 \) \( \bigl[1\) , \( 1\) , \( a^{4} + a^{3} - 4 a^{2} - 2 a + 2\) , \( -2 a^{4} + a^{3} + 8 a^{2} - 2 a - 7\) , \( -a^{3} - a^{2} + 2 a + 2\bigr] \) ${y}^2+{x}{y}+\left(a^{4}+a^{3}-4a^{2}-2a+2\right){y}={x}^{3}+{x}^{2}+\left(-2a^{4}+a^{3}+8a^{2}-2a-7\right){x}-a^{3}-a^{2}+2a+2$
64.1-a3 64.1-a \(\Q(\zeta_{13})^+\) \( 2^{6} \) $0$ $\mathsf{trivial}$ $1$ \( -\frac{1680914269}{32768} \) \( \bigl[a^{5} + a^{4} - 4 a^{3} - 3 a^{2} + 2 a\) , \( -a^{5} + a^{4} + 4 a^{3} - 4 a^{2} - 2 a + 1\) , \( a^{5} + a^{4} - 5 a^{3} - 3 a^{2} + 5 a + 1\) , \( 297 a^{5} - 225 a^{4} - 1336 a^{3} + 923 a^{2} + 1137 a - 917\) , \( 4026 a^{5} - 3233 a^{4} - 18380 a^{3} + 13142 a^{2} + 16483 a - 12186\bigr] \) ${y}^2+\left(a^{5}+a^{4}-4a^{3}-3a^{2}+2a\right){x}{y}+\left(a^{5}+a^{4}-5a^{3}-3a^{2}+5a+1\right){y}={x}^{3}+\left(-a^{5}+a^{4}+4a^{3}-4a^{2}-2a+1\right){x}^{2}+\left(297a^{5}-225a^{4}-1336a^{3}+923a^{2}+1137a-917\right){x}+4026a^{5}-3233a^{4}-18380a^{3}+13142a^{2}+16483a-12186$
64.1-a4 64.1-a \(\Q(\zeta_{13})^+\) \( 2^{6} \) $0$ $\Z/5\Z$ $1$ \( \frac{1331}{8} \) \( \bigl[a^{4} + a^{3} - 3 a^{2} - 2 a\) , \( a^{5} - a^{4} - 5 a^{3} + 4 a^{2} + 6 a - 1\) , \( a^{5} - 5 a^{3} + 5 a + 1\) , \( 11 a^{5} + a^{4} - 51 a^{3} - 5 a^{2} + 55 a + 15\) , \( 49 a^{5} - 8 a^{4} - 244 a^{3} + 3 a^{2} + 281 a + 64\bigr] \) ${y}^2+\left(a^{4}+a^{3}-3a^{2}-2a\right){x}{y}+\left(a^{5}-5a^{3}+5a+1\right){y}={x}^{3}+\left(a^{5}-a^{4}-5a^{3}+4a^{2}+6a-1\right){x}^{2}+\left(11a^{5}+a^{4}-51a^{3}-5a^{2}+55a+15\right){x}+49a^{5}-8a^{4}-244a^{3}+3a^{2}+281a+64$
64.1-a5 64.1-a \(\Q(\zeta_{13})^+\) \( 2^{6} \) $0$ $\Z/5\Z$ $1$ \( \frac{461373}{2} a^{4} - \frac{461373}{2} a^{3} - 922746 a^{2} + 461373 a + \frac{321323}{2} \) \( \bigl[1\) , \( 1\) , \( a^{4} + a^{3} - 4 a^{2} - 2 a + 3\) , \( a^{4} - 2 a^{3} - 4 a^{2} + 4 a\) , \( -2 a^{4} + 7 a^{2} - 3\bigr] \) ${y}^2+{x}{y}+\left(a^{4}+a^{3}-4a^{2}-2a+3\right){y}={x}^{3}+{x}^{2}+\left(a^{4}-2a^{3}-4a^{2}+4a\right){x}-2a^{4}+7a^{2}-3$
64.1-a6 64.1-a \(\Q(\zeta_{13})^+\) \( 2^{6} \) $0$ $\mathsf{trivial}$ $1$ \( \frac{1250637664527933}{32} a^{4} - \frac{1250637664527933}{32} a^{3} - \frac{1250637664527933}{8} a^{2} + \frac{1250637664527933}{16} a + \frac{871975048120043}{32} \) \( \bigl[1\) , \( 1\) , \( a^{4} + a^{3} - 4 a^{2} - 2 a + 2\) , \( 28 a^{4} - 29 a^{3} - 112 a^{2} + 58 a + 58\) , \( 51 a^{4} - 52 a^{3} - 205 a^{2} + 104 a - 3\bigr] \) ${y}^2+{x}{y}+\left(a^{4}+a^{3}-4a^{2}-2a+2\right){y}={x}^{3}+{x}^{2}+\left(28a^{4}-29a^{3}-112a^{2}+58a+58\right){x}+51a^{4}-52a^{3}-205a^{2}+104a-3$
64.1-b1 64.1-b \(\Q(\zeta_{13})^+\) \( 2^{6} \) $1$ $\mathsf{trivial}$ $1.188700458$ \( -\frac{38575685889}{16384} \) \( \bigl[a^{5} + a^{4} - 4 a^{3} - 3 a^{2} + 3 a\) , \( a^{5} + a^{4} - 5 a^{3} - 4 a^{2} + 5 a + 2\) , \( a^{5} + a^{4} - 5 a^{3} - 3 a^{2} + 5 a + 1\) , \( 1801 a^{5} + 962 a^{4} - 7561 a^{3} - 4471 a^{2} + 4023 a + 1051\) , \( 70970 a^{5} + 36003 a^{4} - 301077 a^{3} - 170680 a^{2} + 170108 a + 46396\bigr] \) ${y}^2+\left(a^{5}+a^{4}-4a^{3}-3a^{2}+3a\right){x}{y}+\left(a^{5}+a^{4}-5a^{3}-3a^{2}+5a+1\right){y}={x}^{3}+\left(a^{5}+a^{4}-5a^{3}-4a^{2}+5a+2\right){x}^{2}+\left(1801a^{5}+962a^{4}-7561a^{3}-4471a^{2}+4023a+1051\right){x}+70970a^{5}+36003a^{4}-301077a^{3}-170680a^{2}+170108a+46396$
64.1-b2 64.1-b \(\Q(\zeta_{13})^+\) \( 2^{6} \) $1$ $\Z/7\Z$ $0.169814351$ \( \frac{351}{4} \) \( \bigl[a^{5} + a^{4} - 4 a^{3} - 3 a^{2} + 3 a\) , \( a^{5} + a^{4} - 5 a^{3} - 4 a^{2} + 5 a + 2\) , \( a^{5} + a^{4} - 5 a^{3} - 3 a^{2} + 5 a + 1\) , \( a^{5} + 2 a^{4} - a^{3} - a^{2} + 3 a + 1\) , \( -20 a^{5} - 7 a^{4} + 93 a^{3} + 50 a^{2} - 52 a - 14\bigr] \) ${y}^2+\left(a^{5}+a^{4}-4a^{3}-3a^{2}+3a\right){x}{y}+\left(a^{5}+a^{4}-5a^{3}-3a^{2}+5a+1\right){y}={x}^{3}+\left(a^{5}+a^{4}-5a^{3}-4a^{2}+5a+2\right){x}^{2}+\left(a^{5}+2a^{4}-a^{3}-a^{2}+3a+1\right){x}-20a^{5}-7a^{4}+93a^{3}+50a^{2}-52a-14$
79.1-a1 79.1-a \(\Q(\zeta_{13})^+\) \( 79 \) $1$ $\Z/2\Z$ $0.038527218$ \( \frac{1061663819573662758610}{6241} a^{5} - \frac{2267851397132794939683}{6241} a^{4} - \frac{2731746238796018458192}{6241} a^{3} + \frac{7350272748843635563380}{6241} a^{2} - \frac{1980878737512083984999}{6241} a - \frac{934456702058730400830}{6241} \) \( \bigl[a^{4} - 4 a^{2} + a + 3\) , \( a^{3} + a^{2} - 4 a - 3\) , \( a^{5} - 5 a^{3} + 5 a + 1\) , \( -4 a^{5} - 4 a^{4} + 11 a^{3} + 9 a^{2} + 4 a - 9\) , \( -5 a^{5} - 12 a^{4} + 28 a^{3} + 22 a^{2} - 34 a + 4\bigr] \) ${y}^2+\left(a^{4}-4a^{2}+a+3\right){x}{y}+\left(a^{5}-5a^{3}+5a+1\right){y}={x}^{3}+\left(a^{3}+a^{2}-4a-3\right){x}^{2}+\left(-4a^{5}-4a^{4}+11a^{3}+9a^{2}+4a-9\right){x}-5a^{5}-12a^{4}+28a^{3}+22a^{2}-34a+4$
79.1-a2 79.1-a \(\Q(\zeta_{13})^+\) \( 79 \) $1$ $\Z/2\Z$ $0.019263609$ \( \frac{11177485547}{79} a^{5} - \frac{23876631104}{79} a^{4} - \frac{28760555737}{79} a^{3} + \frac{77385951919}{79} a^{2} - \frac{20855199087}{79} a - \frac{9838223470}{79} \) \( \bigl[a^{4} - 4 a^{2} + a + 3\) , \( a^{3} + a^{2} - 4 a - 3\) , \( a^{5} - 5 a^{3} + 5 a + 1\) , \( a^{5} + a^{4} - 4 a^{3} - a^{2} + 4 a + 1\) , \( a^{5} + 2 a^{4} - a^{3} - 5 a^{2} - 3 a\bigr] \) ${y}^2+\left(a^{4}-4a^{2}+a+3\right){x}{y}+\left(a^{5}-5a^{3}+5a+1\right){y}={x}^{3}+\left(a^{3}+a^{2}-4a-3\right){x}^{2}+\left(a^{5}+a^{4}-4a^{3}-a^{2}+4a+1\right){x}+a^{5}+2a^{4}-a^{3}-5a^{2}-3a$
79.1-b1 79.1-b \(\Q(\zeta_{13})^+\) \( 79 \) $1$ $\mathsf{trivial}$ $0.001171369$ \( \frac{10278832194}{6241} a^{5} - \frac{29929897535}{6241} a^{4} + \frac{238271915}{6241} a^{3} + \frac{45269862214}{6241} a^{2} - \frac{15374036487}{6241} a - \frac{6225749444}{6241} \) \( \bigl[a^{4} - 4 a^{2} + 2\) , \( a^{5} - 6 a^{3} + 9 a + 1\) , \( a^{4} + a^{3} - 4 a^{2} - 3 a + 3\) , \( 9 a^{5} - 40 a^{3} - 5 a^{2} + 31 a + 3\) , \( -7 a^{5} - 5 a^{4} + 28 a^{3} + 22 a^{2} - 11 a - 5\bigr] \) ${y}^2+\left(a^{4}-4a^{2}+2\right){x}{y}+\left(a^{4}+a^{3}-4a^{2}-3a+3\right){y}={x}^{3}+\left(a^{5}-6a^{3}+9a+1\right){x}^{2}+\left(9a^{5}-40a^{3}-5a^{2}+31a+3\right){x}-7a^{5}-5a^{4}+28a^{3}+22a^{2}-11a-5$
79.1-c1 79.1-c \(\Q(\zeta_{13})^+\) \( 79 \) $0$ $\Z/2\Z$ $1$ \( \frac{55023306124512922647620}{3077056399} a^{5} - \frac{118530438657502667153870}{3077056399} a^{4} - \frac{141076757522153256512802}{3077056399} a^{3} + \frac{384280328755279198355509}{3077056399} a^{2} - \frac{104160003334531508958775}{3077056399} a - \frac{48974567195018206908192}{3077056399} \) \( \bigl[a^{4} - 3 a^{2} + a\) , \( -a^{4} + 3 a^{2} + a - 1\) , \( a^{4} - 3 a^{2} + a\) , \( -36 a^{5} + 65 a^{4} + 103 a^{3} - 230 a^{2} + 50 a + 24\) , \( -259 a^{5} + 514 a^{4} + 687 a^{3} - 1743 a^{2} + 502 a + 177\bigr] \) ${y}^2+\left(a^{4}-3a^{2}+a\right){x}{y}+\left(a^{4}-3a^{2}+a\right){y}={x}^{3}+\left(-a^{4}+3a^{2}+a-1\right){x}^{2}+\left(-36a^{5}+65a^{4}+103a^{3}-230a^{2}+50a+24\right){x}-259a^{5}+514a^{4}+687a^{3}-1743a^{2}+502a+177$
79.1-c2 79.1-c \(\Q(\zeta_{13})^+\) \( 79 \) $0$ $\Z/10\Z$ $1$ \( \frac{12273524}{79} a^{5} - \frac{3709875}{79} a^{4} - \frac{63845719}{79} a^{3} + \frac{4394312}{79} a^{2} + \frac{76199147}{79} a + \frac{16823747}{79} \) \( \bigl[a^{5} - 4 a^{3} + 3 a + 1\) , \( a^{5} + a^{4} - 4 a^{3} - 3 a^{2} + a - 1\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -12 a^{5} + 3 a^{4} + 61 a^{3} - 66 a - 15\) , \( 37 a^{5} - 7 a^{4} - 188 a^{3} + a^{2} + 217 a + 50\bigr] \) ${y}^2+\left(a^{5}-4a^{3}+3a+1\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{5}+a^{4}-4a^{3}-3a^{2}+a-1\right){x}^{2}+\left(-12a^{5}+3a^{4}+61a^{3}-66a-15\right){x}+37a^{5}-7a^{4}-188a^{3}+a^{2}+217a+50$
79.1-c3 79.1-c \(\Q(\zeta_{13})^+\) \( 79 \) $0$ $\Z/10\Z$ $1$ \( \frac{628049103430807}{6241} a^{5} - \frac{182589231311538}{6241} a^{4} - \frac{3269795114557063}{6241} a^{3} + \frac{193031548929585}{6241} a^{2} + \frac{3905423405681707}{6241} a + \frac{885817850043621}{6241} \) \( \bigl[a^{4} - 3 a^{2} + a\) , \( -a^{4} + 3 a^{2} + a - 1\) , \( a^{4} - 3 a^{2} + a\) , \( 4 a^{5} - 10 a^{4} - 12 a^{3} + 30 a^{2} + 10 a - 21\) , \( -15 a^{5} + 27 a^{4} + 43 a^{3} - 71 a^{2} - 34 a + 43\bigr] \) ${y}^2+\left(a^{4}-3a^{2}+a\right){x}{y}+\left(a^{4}-3a^{2}+a\right){y}={x}^{3}+\left(-a^{4}+3a^{2}+a-1\right){x}^{2}+\left(4a^{5}-10a^{4}-12a^{3}+30a^{2}+10a-21\right){x}-15a^{5}+27a^{4}+43a^{3}-71a^{2}-34a+43$
79.1-c4 79.1-c \(\Q(\zeta_{13})^+\) \( 79 \) $0$ $\Z/2\Z$ $1$ \( \frac{127794683808852859853832701635933437922415}{9468276082626847201} a^{5} + \frac{120367721300657709905855839586570070913343}{9468276082626847201} a^{4} - \frac{405233310886078181372607580308878671807916}{9468276082626847201} a^{3} - \frac{275737199471198880155191606074376860275461}{9468276082626847201} a^{2} + \frac{231318547680046130469307232906240352482422}{9468276082626847201} a + \frac{65809650751079058294984275680386783414465}{9468276082626847201} \) \( \bigl[a^{4} - 3 a^{2} + a\) , \( -a^{4} + 3 a^{2} + a - 1\) , \( a^{4} - 3 a^{2} + a\) , \( 4 a^{5} - 200 a^{4} + 73 a^{3} + 365 a^{2} + 85 a - 231\) , \( 24 a^{5} - 2877 a^{4} + 1997 a^{3} + 4755 a^{2} - 931 a - 1732\bigr] \) ${y}^2+\left(a^{4}-3a^{2}+a\right){x}{y}+\left(a^{4}-3a^{2}+a\right){y}={x}^{3}+\left(-a^{4}+3a^{2}+a-1\right){x}^{2}+\left(4a^{5}-200a^{4}+73a^{3}+365a^{2}+85a-231\right){x}+24a^{5}-2877a^{4}+1997a^{3}+4755a^{2}-931a-1732$
79.1-d1 79.1-d \(\Q(\zeta_{13})^+\) \( 79 \) $0 \le r \le 2$ $\Z/2\Z$ $1$ \( -\frac{2762735175183868847277552646275174}{493039} a^{5} - \frac{1373138770767122616960080619238639}{493039} a^{4} + \frac{11758057618620054164219573960351571}{493039} a^{3} + \frac{6551124309568948704114188838899166}{493039} a^{2} - \frac{6769237134468529246169518671845202}{493039} a - \frac{1845487978585672096436224846805198}{493039} \) \( \bigl[a^{5} - 5 a^{3} + 6 a\) , \( a^{5} + a^{4} - 4 a^{3} - 4 a^{2} + 2 a + 2\) , \( a^{2} + a - 1\) , \( -50 a^{5} - 47 a^{4} + 212 a^{3} + 148 a^{2} - 212 a - 107\) , \( -18 a^{5} + 417 a^{4} + 354 a^{3} - 1064 a^{2} - 530 a + 27\bigr] \) ${y}^2+\left(a^{5}-5a^{3}+6a\right){x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+\left(a^{5}+a^{4}-4a^{3}-4a^{2}+2a+2\right){x}^{2}+\left(-50a^{5}-47a^{4}+212a^{3}+148a^{2}-212a-107\right){x}-18a^{5}+417a^{4}+354a^{3}-1064a^{2}-530a+27$
79.1-d2 79.1-d \(\Q(\zeta_{13})^+\) \( 79 \) $0 \le r \le 2$ $\Z/2\Z\oplus\Z/2\Z$ $1$ \( -\frac{7947972766113080746071972}{243087455521} a^{5} - \frac{3950313317752480802186229}{243087455521} a^{4} + \frac{33826159878792106034273209}{243087455521} a^{3} + \frac{18846597413196990291447308}{243087455521} a^{2} - \frac{19474075142746010748953494}{243087455521} a - \frac{5309190805526992370823771}{243087455521} \) \( \bigl[a^{5} - 5 a^{3} + a^{2} + 6 a - 1\) , \( a^{5} + a^{4} - 4 a^{3} - 4 a^{2} + a + 2\) , \( a^{4} + a^{3} - 4 a^{2} - 3 a + 2\) , \( -45 a^{5} + 8 a^{4} + 237 a^{3} + 9 a^{2} - 282 a - 82\) , \( 78 a^{5} - 32 a^{4} - 397 a^{3} + 76 a^{2} + 458 a + 41\bigr] \) ${y}^2+\left(a^{5}-5a^{3}+a^{2}+6a-1\right){x}{y}+\left(a^{4}+a^{3}-4a^{2}-3a+2\right){y}={x}^{3}+\left(a^{5}+a^{4}-4a^{3}-4a^{2}+a+2\right){x}^{2}+\left(-45a^{5}+8a^{4}+237a^{3}+9a^{2}-282a-82\right){x}+78a^{5}-32a^{4}-397a^{3}+76a^{2}+458a+41$
79.1-d3 79.1-d \(\Q(\zeta_{13})^+\) \( 79 \) $0 \le r \le 2$ $\Z/4\Z$ $1$ \( \frac{229222312512}{79} a^{5} - \frac{489659661913}{79} a^{4} - \frac{589830121776}{79} a^{3} + \frac{1587030264768}{79} a^{2} - \frac{427631180184}{79} a - \frac{201808382600}{79} \) \( \bigl[a^{5} + a^{4} - 4 a^{3} - 3 a^{2} + 2 a\) , \( a^{5} - 5 a^{3} - a^{2} + 4 a + 2\) , \( a^{5} - 4 a^{3} + 2 a + 1\) , \( 10 a^{5} - 5 a^{4} - 46 a^{3} + 18 a^{2} + 40 a - 18\) , \( 12 a^{5} - 16 a^{4} - 57 a^{3} + 66 a^{2} + 60 a - 53\bigr] \) ${y}^2+\left(a^{5}+a^{4}-4a^{3}-3a^{2}+2a\right){x}{y}+\left(a^{5}-4a^{3}+2a+1\right){y}={x}^{3}+\left(a^{5}-5a^{3}-a^{2}+4a+2\right){x}^{2}+\left(10a^{5}-5a^{4}-46a^{3}+18a^{2}+40a-18\right){x}+12a^{5}-16a^{4}-57a^{3}+66a^{2}+60a-53$
79.1-d4 79.1-d \(\Q(\zeta_{13})^+\) \( 79 \) $0 \le r \le 2$ $\Z/4\Z$ $1$ \( -\frac{1134795849368}{493039} a^{5} - \frac{640510493225}{493039} a^{4} + \frac{4738281412113}{493039} a^{3} + \frac{2951102426900}{493039} a^{2} - \frac{2417902662226}{493039} a - \frac{686687764294}{493039} \) \( \bigl[a^{5} - 4 a^{3} + 2 a + 1\) , \( a^{5} + a^{4} - 4 a^{3} - 4 a^{2} + 2 a + 1\) , \( a^{5} + a^{4} - 4 a^{3} - 4 a^{2} + 3 a + 3\) , \( 47 a^{5} + 24 a^{4} - 200 a^{3} - 114 a^{2} + 114 a + 32\) , \( 221 a^{5} + 109 a^{4} - 941 a^{3} - 521 a^{2} + 542 a + 145\bigr] \) ${y}^2+\left(a^{5}-4a^{3}+2a+1\right){x}{y}+\left(a^{5}+a^{4}-4a^{3}-4a^{2}+3a+3\right){y}={x}^{3}+\left(a^{5}+a^{4}-4a^{3}-4a^{2}+2a+1\right){x}^{2}+\left(47a^{5}+24a^{4}-200a^{3}-114a^{2}+114a+32\right){x}+221a^{5}+109a^{4}-941a^{3}-521a^{2}+542a+145$
79.1-d5 79.1-d \(\Q(\zeta_{13})^+\) \( 79 \) $0 \le r \le 2$ $\Z/2\Z$ $1$ \( \frac{2033369859844863642437231380090}{59091511031674153381441} a^{5} - \frac{4735260214072170447973367488703}{59091511031674153381441} a^{4} - \frac{3625278097962318215004518456861}{59091511031674153381441} a^{3} + \frac{14266749716321853993630284563838}{59091511031674153381441} a^{2} - \frac{8936329652263048501846210029746}{59091511031674153381441} a + \frac{1288287690387077768870911413346}{59091511031674153381441} \) \( \bigl[a^{5} - 5 a^{3} + a^{2} + 5 a - 2\) , \( a^{3} + a^{2} - 3 a - 3\) , \( a^{3} - 3 a + 1\) , \( 97 a^{5} + 9 a^{4} - 446 a^{3} - 97 a^{2} + 388 a + 71\) , \( 349 a^{5} + 145 a^{4} - 1477 a^{3} - 737 a^{2} + 879 a + 185\bigr] \) ${y}^2+\left(a^{5}-5a^{3}+a^{2}+5a-2\right){x}{y}+\left(a^{3}-3a+1\right){y}={x}^{3}+\left(a^{3}+a^{2}-3a-3\right){x}^{2}+\left(97a^{5}+9a^{4}-446a^{3}-97a^{2}+388a+71\right){x}+349a^{5}+145a^{4}-1477a^{3}-737a^{2}+879a+185$
79.1-d6 79.1-d \(\Q(\zeta_{13})^+\) \( 79 \) $0 \le r \le 2$ $\Z/2\Z\oplus\Z/2\Z$ $1$ \( -\frac{51317863475416}{6241} a^{5} + \frac{1667446511596884}{6241} a^{4} + \frac{1365100078844124}{6241} a^{3} - \frac{5619927792030191}{6241} a^{2} - \frac{2413322991143088}{6241} a + \frac{3638886803263608}{6241} \) \( \bigl[a^{4} + a^{3} - 3 a^{2} - 2 a\) , \( a^{5} - a^{4} - 6 a^{3} + 5 a^{2} + 9 a - 3\) , \( a^{3} - 3 a\) , \( -14 a^{5} + 23 a^{4} + 53 a^{3} - 63 a^{2} - 16 a - 18\) , \( 9 a^{5} - 89 a^{4} + 90 a^{3} + 317 a^{2} - 346 a - 114\bigr] \) ${y}^2+\left(a^{4}+a^{3}-3a^{2}-2a\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(a^{5}-a^{4}-6a^{3}+5a^{2}+9a-3\right){x}^{2}+\left(-14a^{5}+23a^{4}+53a^{3}-63a^{2}-16a-18\right){x}+9a^{5}-89a^{4}+90a^{3}+317a^{2}-346a-114$
79.1-d7 79.1-d \(\Q(\zeta_{13})^+\) \( 79 \) $0 \le r \le 2$ $\Z/2\Z$ $1$ \( \frac{14975057382763351033927532}{79} a^{5} + \frac{14104761450659533335839904}{79} a^{4} - \frac{47485481470815573950189484}{79} a^{3} - \frac{32311049678505073476173738}{79} a^{2} + \frac{27106045692032720097856969}{79} a + \frac{7711614205444837873329076}{79} \) \( \bigl[a^{4} + a^{3} - 3 a^{2} - 2 a\) , \( a^{5} - a^{4} - 6 a^{3} + 5 a^{2} + 9 a - 3\) , \( a^{3} - 3 a\) , \( -174 a^{5} + 48 a^{4} + 888 a^{3} - 53 a^{2} - 1036 a - 253\) , \( -1944 a^{5} + 404 a^{4} + 10398 a^{3} - 6 a^{2} - 12923 a - 2986\bigr] \) ${y}^2+\left(a^{4}+a^{3}-3a^{2}-2a\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(a^{5}-a^{4}-6a^{3}+5a^{2}+9a-3\right){x}^{2}+\left(-174a^{5}+48a^{4}+888a^{3}-53a^{2}-1036a-253\right){x}-1944a^{5}+404a^{4}+10398a^{3}-6a^{2}-12923a-2986$
79.1-d8 79.1-d \(\Q(\zeta_{13})^+\) \( 79 \) $0 \le r \le 2$ $\Z/2\Z$ $1$ \( -\frac{3050233510862210151986369219028}{38950081} a^{5} + \frac{3785563553688770366539690135200}{38950081} a^{4} + \frac{14338569026938982489863055482260}{38950081} a^{3} - \frac{15657581063688821579016146021962}{38950081} a^{2} - \frac{14526775380595228987077436189607}{38950081} a + \frac{12652719090941410050519681882308}{38950081} \) \( \bigl[a^{5} - 5 a^{3} + a^{2} + 6 a - 2\) , \( a^{5} + a^{4} - 6 a^{3} - 4 a^{2} + 7 a + 2\) , \( a^{5} + a^{4} - 4 a^{3} - 4 a^{2} + 2 a + 2\) , \( -269 a^{5} + 97 a^{4} + 1409 a^{3} - 146 a^{2} - 1715 a - 393\) , \( 70265 a^{5} + 38655 a^{4} - 294531 a^{3} - 179297 a^{2} + 154272 a + 43383\bigr] \) ${y}^2+\left(a^{5}-5a^{3}+a^{2}+6a-2\right){x}{y}+\left(a^{5}+a^{4}-4a^{3}-4a^{2}+2a+2\right){y}={x}^{3}+\left(a^{5}+a^{4}-6a^{3}-4a^{2}+7a+2\right){x}^{2}+\left(-269a^{5}+97a^{4}+1409a^{3}-146a^{2}-1715a-393\right){x}+70265a^{5}+38655a^{4}-294531a^{3}-179297a^{2}+154272a+43383$
79.2-a1 79.2-a \(\Q(\zeta_{13})^+\) \( 79 \) $1$ $\Z/2\Z$ $0.019263609$ \( -\frac{19238641932}{79} a^{5} + \frac{23876631104}{79} a^{4} + \frac{90437151053}{79} a^{3} - \frac{98756765310}{79} a^{2} - \frac{91624179396}{79} a + \frac{79804268844}{79} \) \( \bigl[a^{5} - 4 a^{3} + a^{2} + 2 a - 2\) , \( a^{4} - 5 a^{2} + 4\) , \( a^{4} - 4 a^{2} + 2\) , \( -a^{5} + 3 a^{4} + 4 a^{3} - 12 a^{2} - 3 a + 8\) , \( 5 a^{5} - 10 a^{4} - 13 a^{3} + 32 a^{2} - 9 a - 3\bigr] \) ${y}^2+\left(a^{5}-4a^{3}+a^{2}+2a-2\right){x}{y}+\left(a^{4}-4a^{2}+2\right){y}={x}^{3}+\left(a^{4}-5a^{2}+4\right){x}^{2}+\left(-a^{5}+3a^{4}+4a^{3}-12a^{2}-3a+8\right){x}+5a^{5}-10a^{4}-13a^{3}+32a^{2}-9a-3$
79.2-a2 79.2-a \(\Q(\zeta_{13})^+\) \( 79 \) $1$ $\Z/2\Z$ $0.038527218$ \( -\frac{1827330655296306713158}{6241} a^{5} + \frac{2267851397132794939683}{6241} a^{4} + \frac{8589934719036282821459}{6241} a^{3} - \frac{9380127050470680153907}{6241} a^{2} - \frac{8702685181704913513870}{6241} a + \frac{7579977528109156697231}{6241} \) \( \bigl[a^{5} - 4 a^{3} + a^{2} + 2 a - 2\) , \( a^{4} - 5 a^{2} + 4\) , \( a^{4} - 4 a^{2} + 2\) , \( -16 a^{5} + 43 a^{4} + 24 a^{3} - 132 a^{2} + 82 a - 7\) , \( 127 a^{5} - 279 a^{4} - 308 a^{3} + 894 a^{2} - 296 a - 82\bigr] \) ${y}^2+\left(a^{5}-4a^{3}+a^{2}+2a-2\right){x}{y}+\left(a^{4}-4a^{2}+2\right){y}={x}^{3}+\left(a^{4}-5a^{2}+4\right){x}^{2}+\left(-16a^{5}+43a^{4}+24a^{3}-132a^{2}+82a-7\right){x}+127a^{5}-279a^{4}-308a^{3}+894a^{2}-296a-82$
79.2-b1 79.2-b \(\Q(\zeta_{13})^+\) \( 79 \) $1$ $\mathsf{trivial}$ $0.001171369$ \( \frac{58199203663}{6241} a^{5} + \frac{29929897535}{6241} a^{4} - \frac{246476187924}{6241} a^{3} - \frac{141422125490}{6241} a^{2} + \frac{137785461801}{6241} a + \frac{37929353079}{6241} \) \( \bigl[a^{5} + a^{4} - 5 a^{3} - 3 a^{2} + 6 a\) , \( a^{4} - 3 a^{2} + a - 1\) , \( a^{4} + a^{3} - 4 a^{2} - 2 a + 3\) , \( 9 a^{5} + 7 a^{4} - 35 a^{3} - 27 a^{2} + 16 a + 8\) , \( -16 a^{5} - 8 a^{4} + 71 a^{3} + 45 a^{2} - 37 a - 13\bigr] \) ${y}^2+\left(a^{5}+a^{4}-5a^{3}-3a^{2}+6a\right){x}{y}+\left(a^{4}+a^{3}-4a^{2}-2a+3\right){y}={x}^{3}+\left(a^{4}-3a^{2}+a-1\right){x}^{2}+\left(9a^{5}+7a^{4}-35a^{3}-27a^{2}+16a+8\right){x}-16a^{5}-8a^{4}+71a^{3}+45a^{2}-37a-13$
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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.