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Label Class Base field Conductor norm Rank Torsion CM Regulator Period Leading coeff j-invariant Weierstrass coefficients Weierstrass equation
25.1-a1 25.1-a \(\Q(\zeta_{15})^+\) \( 5^{2} \) $0$ $\Z/2\Z$ $-15$ $1$ $49.92252905$ 0.744201123 \( -85995 a^{3} + 257985 a - 138510 \) \( \bigl[a^{2} + a - 2\) , \( -a^{3} - a^{2} + 4 a + 2\) , \( a^{3} - 2 a + 1\) , \( -5 a^{3} - 4 a^{2} + 13 a + 6\) , \( -7 a^{3} - 6 a^{2} + 19 a + 4\bigr] \) ${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{3}-2a+1\right){y}={x}^{3}+\left(-a^{3}-a^{2}+4a+2\right){x}^{2}+\left(-5a^{3}-4a^{2}+13a+6\right){x}-7a^{3}-6a^{2}+19a+4$
25.1-a2 25.1-a \(\Q(\zeta_{15})^+\) \( 5^{2} \) $0$ $\Z/2\Z$ $-60$ $1$ $49.92252905$ 0.744201123 \( 16554983445 a^{3} - 49664950335 a + 26786530035 \) \( \bigl[a^{2} + a - 2\) , \( -a^{3} - a^{2} + 4 a + 2\) , \( a^{3} - 2 a + 1\) , \( -45 a^{3} - 39 a^{2} + 113 a + 26\) , \( -258 a^{3} - 214 a^{2} + 641 a + 138\bigr] \) ${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{3}-2a+1\right){y}={x}^{3}+\left(-a^{3}-a^{2}+4a+2\right){x}^{2}+\left(-45a^{3}-39a^{2}+113a+26\right){x}-258a^{3}-214a^{2}+641a+138$
25.1-a3 25.1-a \(\Q(\zeta_{15})^+\) \( 5^{2} \) $0$ $\Z/10\Z$ $-15$ $1$ $1248.063226$ 0.744201123 \( -85995 a^{3} + 257985 a - 138510 \) \( \bigl[a^{2} - 1\) , \( a - 1\) , \( a^{3} + a^{2} - 2 a - 1\) , \( -716912 a^{3} - 592951 a^{2} + 1784272 a + 392379\) , \( 454902270 a^{3} + 376245535 a^{2} - 1132174281 a - 248976264\bigr] \) ${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-716912a^{3}-592951a^{2}+1784272a+392379\right){x}+454902270a^{3}+376245535a^{2}-1132174281a-248976264$
25.1-a4 25.1-a \(\Q(\zeta_{15})^+\) \( 5^{2} \) $0$ $\Z/10\Z$ $-60$ $1$ $1248.063226$ 0.744201123 \( 16554983445 a^{3} - 49664950335 a + 26786530035 \) \( \bigl[a^{2} - 1\) , \( a - 1\) , \( a^{3} + a^{2} - 2 a - 1\) , \( -11484892 a^{3} - 9499051 a^{2} + 28583937 a + 6285889\) , \( 29062760688 a^{3} + 24037545339 a^{2} - 72332262037 a - 15906576101\bigr] \) ${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-11484892a^{3}-9499051a^{2}+28583937a+6285889\right){x}+29062760688a^{3}+24037545339a^{2}-72332262037a-15906576101$
25.1-a5 25.1-a \(\Q(\zeta_{15})^+\) \( 5^{2} \) $0$ $\Z/10\Z$ $-15$ $1$ $1248.063226$ 0.744201123 \( 85995 a^{3} - 257985 a - 52515 \) \( \bigl[a^{3} - 2 a\) , \( -a^{3} + a^{2} + 3 a - 3\) , \( a + 1\) , \( 5 a^{3} + 4 a^{2} - 15 a + 1\) , \( 14 a^{3} + 13 a^{2} - 36 a - 10\bigr] \) ${y}^2+\left(a^{3}-2a\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-3\right){x}^{2}+\left(5a^{3}+4a^{2}-15a+1\right){x}+14a^{3}+13a^{2}-36a-10$
25.1-a6 25.1-a \(\Q(\zeta_{15})^+\) \( 5^{2} \) $0$ $\Z/10\Z$ $-60$ $1$ $1248.063226$ 0.744201123 \( -16554983445 a^{3} + 49664950335 a + 10231546590 \) \( \bigl[a^{3} - 2 a\) , \( -a^{3} + a^{2} + 3 a - 3\) , \( a + 1\) , \( -35 a^{3} - 31 a^{2} + 85 a + 21\) , \( 110 a^{3} + 91 a^{2} - 273 a - 59\bigr] \) ${y}^2+\left(a^{3}-2a\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-3\right){x}^{2}+\left(-35a^{3}-31a^{2}+85a+21\right){x}+110a^{3}+91a^{2}-273a-59$
25.1-a7 25.1-a \(\Q(\zeta_{15})^+\) \( 5^{2} \) $0$ $\Z/2\Z$ $-15$ $1$ $49.92252905$ 0.744201123 \( 85995 a^{3} - 257985 a - 52515 \) \( \bigl[a^{3} - 2 a + 1\) , \( a^{3} + a^{2} - 4 a - 3\) , \( a^{2} + a - 1\) , \( 29593 a^{3} + 24472 a^{2} - 73662 a - 16197\) , \( -4979614 a^{3} - 4118597 a^{2} + 12393403 a + 2725431\bigr] \) ${y}^2+\left(a^{3}-2a+1\right){x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+\left(a^{3}+a^{2}-4a-3\right){x}^{2}+\left(29593a^{3}+24472a^{2}-73662a-16197\right){x}-4979614a^{3}-4118597a^{2}+12393403a+2725431$
25.1-a8 25.1-a \(\Q(\zeta_{15})^+\) \( 5^{2} \) $0$ $\Z/2\Z$ $-60$ $1$ $49.92252905$ 0.744201123 \( -16554983445 a^{3} + 49664950335 a + 10231546590 \) \( \bigl[a^{3} - 2 a + 1\) , \( a^{3} + a^{2} - 4 a - 3\) , \( a^{2} + a - 1\) , \( -199462 a^{3} - 165053 a^{2} + 496253 a + 109138\) , \( -50977357 a^{3} - 42163431 a^{2} + 126872829 a + 27900618\bigr] \) ${y}^2+\left(a^{3}-2a+1\right){x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+\left(a^{3}+a^{2}-4a-3\right){x}^{2}+\left(-199462a^{3}-165053a^{2}+496253a+109138\right){x}-50977357a^{3}-42163431a^{2}+126872829a+27900618$
25.1-b1 25.1-b \(\Q(\zeta_{15})^+\) \( 5^{2} \) $0$ $\Z/5\Z$ $-3$ $1$ $603.3662684$ 0.719556262 \( 0 \) \( \bigl[0\) , \( -a^{3} + a^{2} + 3 a - 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -a + 2\) , \( -a^{2} - a + 3\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-2\right){x}^{2}+\left(-a+2\right){x}-a^{2}-a+3$
25.1-b2 25.1-b \(\Q(\zeta_{15})^+\) \( 5^{2} \) $0$ $\Z/5\Z$ $-3$ $1$ $603.3662684$ 0.719556262 \( 0 \) \( \bigl[0\) , \( a^{3} - 2 a + 1\) , \( a^{2} - 1\) , \( a^{3} + a^{2} - 3 a\) , \( -11017 a^{3} - 9112 a^{2} + 27421 a + 6030\bigr] \) ${y}^2+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-2a+1\right){x}^{2}+\left(a^{3}+a^{2}-3a\right){x}-11017a^{3}-9112a^{2}+27421a+6030$
31.1-a1 31.1-a \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/4\Z$ $1$ $226.2402485$ 0.843147624 \( \frac{13262138608154212980818020352}{923521} a^{3} - \frac{16034680544860826249617094257}{923521} a^{2} - \frac{49696393399932989894285277456}{923521} a + \frac{63437929696698697519404008684}{923521} \) \( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( a^{3} - a^{2} - 3 a + 1\) , \( 1\) , \( -427 a^{3} - 156 a^{2} + 1210 a - 328\) , \( 5301 a^{3} + 2573 a^{2} - 14446 a + 2048\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-1\right){x}{y}+{y}={x}^{3}+\left(a^{3}-a^{2}-3a+1\right){x}^{2}+\left(-427a^{3}-156a^{2}+1210a-328\right){x}+5301a^{3}+2573a^{2}-14446a+2048$
31.1-a2 31.1-a \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/2\Z$ $1$ $14.14001553$ 0.843147624 \( -\frac{69643673441152672038181763072}{727423121747185263828481} a^{3} + \frac{84890779633537182060967675121}{727423121747185263828481} a^{2} + \frac{261357927431140745378290491664}{727423121747185263828481} a - \frac{334879646814468933111035546908}{727423121747185263828481} \) \( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( a^{3} - a^{2} - 3 a + 1\) , \( 1\) , \( 113 a^{3} + 114 a^{2} - 280 a - 98\) , \( -a^{3} - 75 a^{2} + 44 a + 78\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-1\right){x}{y}+{y}={x}^{3}+\left(a^{3}-a^{2}-3a+1\right){x}^{2}+\left(113a^{3}+114a^{2}-280a-98\right){x}-a^{3}-75a^{2}+44a+78$
31.1-a3 31.1-a \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $226.2402485$ 0.843147624 \( \frac{43561848213205037510145}{852891037441} a^{3} - \frac{52668771245192606365444}{852891037441} a^{2} - \frac{163236628849912597561348}{852891037441} a + \frac{208373205080817133396719}{852891037441} \) \( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( a^{3} - a^{2} - 3 a + 1\) , \( 1\) , \( -37 a^{3} - 21 a^{2} + 105 a - 13\) , \( 2 a^{3} - 25 a^{2} - 17 a + 63\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-1\right){x}{y}+{y}={x}^{3}+\left(a^{3}-a^{2}-3a+1\right){x}^{2}+\left(-37a^{3}-21a^{2}+105a-13\right){x}+2a^{3}-25a^{2}-17a+63$
31.1-a4 31.1-a \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/4\Z$ $1$ $226.2402485$ 0.843147624 \( -\frac{119292}{961} a^{3} - \frac{1134908}{961} a^{2} + \frac{2242557}{961} a + \frac{309574}{961} \) \( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( a^{3} - a^{2} - 3 a + 1\) , \( 1\) , \( -2 a^{3} - a^{2} + 5 a + 2\) , \( -2 a^{3} - 2 a^{2} + 5 a + 1\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-1\right){x}{y}+{y}={x}^{3}+\left(a^{3}-a^{2}-3a+1\right){x}^{2}+\left(-2a^{3}-a^{2}+5a+2\right){x}-2a^{3}-2a^{2}+5a+1$
31.1-a5 31.1-a \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $226.2402485$ 0.843147624 \( -\frac{2933497938285}{923521} a^{3} + \frac{9133676668039}{923521} a^{2} - \frac{6067822613957}{923521} a - \frac{1204128338708}{923521} \) \( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( a^{3} - a^{2} - 3 a + 1\) , \( 1\) , \( -22 a^{3} - 21 a^{2} + 60 a + 12\) , \( -77 a^{3} - 67 a^{2} + 195 a + 43\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-1\right){x}{y}+{y}={x}^{3}+\left(a^{3}-a^{2}-3a+1\right){x}^{2}+\left(-22a^{3}-21a^{2}+60a+12\right){x}-77a^{3}-67a^{2}+195a+43$
31.1-a6 31.1-a \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/2\Z$ $1$ $14.14001553$ 0.843147624 \( -\frac{236284316384699073}{961} a^{3} + \frac{698632890478410244}{961} a^{2} - \frac{421506652954098940}{961} a - \frac{120811375486877791}{961} \) \( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( a^{3} - a^{2} - 3 a + 1\) , \( 1\) , \( -327 a^{3} - 341 a^{2} + 895 a + 197\) , \( -4716 a^{3} - 4369 a^{2} + 12267 a + 2711\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-1\right){x}{y}+{y}={x}^{3}+\left(a^{3}-a^{2}-3a+1\right){x}^{2}+\left(-327a^{3}-341a^{2}+895a+197\right){x}-4716a^{3}-4369a^{2}+12267a+2711$
31.1-b1 31.1-b \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/2\Z$ $1$ $2.959428063$ 0.705864781 \( \frac{13262138608154212980818020352}{923521} a^{3} - \frac{16034680544860826249617094257}{923521} a^{2} - \frac{49696393399932989894285277456}{923521} a + \frac{63437929696698697519404008684}{923521} \) \( \bigl[a^{2} - 2\) , \( -a^{2} + 1\) , \( 0\) , \( -80 a^{3} + 335 a^{2} + 315 a - 1284\) , \( -2177 a^{3} + 4753 a^{2} + 8378 a - 18117\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(-a^{2}+1\right){x}^{2}+\left(-80a^{3}+335a^{2}+315a-1284\right){x}-2177a^{3}+4753a^{2}+8378a-18117$
31.1-b2 31.1-b \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $47.35084901$ 0.705864781 \( \frac{43561848213205037510145}{852891037441} a^{3} - \frac{52668771245192606365444}{852891037441} a^{2} - \frac{163236628849912597561348}{852891037441} a + \frac{208373205080817133396719}{852891037441} \) \( \bigl[a^{2} - 2\) , \( -a^{2} + 1\) , \( 0\) , \( -5 a^{3} + 20 a^{2} + 20 a - 79\) , \( -42 a^{3} + 85 a^{2} + 161 a - 317\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(-a^{2}+1\right){x}^{2}+\left(-5a^{3}+20a^{2}+20a-79\right){x}-42a^{3}+85a^{2}+161a-317$
31.1-b3 31.1-b \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/2\Z$ $1$ $2.959428063$ 0.705864781 \( -\frac{69643673441152672038181763072}{727423121747185263828481} a^{3} + \frac{84890779633537182060967675121}{727423121747185263828481} a^{2} + \frac{261357927431140745378290491664}{727423121747185263828481} a - \frac{334879646814468933111035546908}{727423121747185263828481} \) \( \bigl[a^{2} - 2\) , \( -a^{2} + 1\) , \( 0\) , \( -10 a^{3} + 25 a^{2} + 45 a - 74\) , \( 9 a^{3} + 101 a^{2} - 28 a - 357\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(-a^{2}+1\right){x}^{2}+\left(-10a^{3}+25a^{2}+45a-74\right){x}+9a^{3}+101a^{2}-28a-357$
31.1-b4 31.1-b \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/8\Z$ $1$ $757.6135842$ 0.705864781 \( -\frac{236284316384699073}{961} a^{3} + \frac{698632890478410244}{961} a^{2} - \frac{421506652954098940}{961} a - \frac{120811375486877791}{961} \) \( \bigl[a^{2} - 2\) , \( -a^{2} + 1\) , \( 0\) , \( 5 a^{3} - 20 a^{2} - 20 a - 9\) , \( -4 a^{3} + 99 a^{2} + 23 a - 5\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(-a^{2}+1\right){x}^{2}+\left(5a^{3}-20a^{2}-20a-9\right){x}-4a^{3}+99a^{2}+23a-5$
31.1-b5 31.1-b \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/2\Z\oplus\Z/4\Z$ $1$ $757.6135842$ 0.705864781 \( -\frac{2933497938285}{923521} a^{3} + \frac{9133676668039}{923521} a^{2} - \frac{6067822613957}{923521} a - \frac{1204128338708}{923521} \) \( \bigl[a^{2} - 2\) , \( -a^{2} + 1\) , \( 0\) , \( -4\) , \( -a^{3} + 4 a^{2} + 4 a - 7\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(-a^{2}+1\right){x}^{2}-4{x}-a^{3}+4a^{2}+4a-7$
31.1-b6 31.1-b \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/8\Z$ $1$ $757.6135842$ 0.705864781 \( -\frac{119292}{961} a^{3} - \frac{1134908}{961} a^{2} + \frac{2242557}{961} a + \frac{309574}{961} \) \( \bigl[a^{2} - 2\) , \( -a^{2} + 1\) , \( 0\) , \( 1\) , \( 0\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(-a^{2}+1\right){x}^{2}+{x}$
31.2-a1 31.2-a \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/2\Z$ $1$ $14.14001553$ 0.843147624 \( \frac{934917206863109317}{961} a^{3} - \frac{1130359602108196159}{961} a^{2} - \frac{3503384511067738195}{961} a + \frac{4472090816549444332}{961} \) \( \bigl[a + 1\) , \( a^{3} - 3 a\) , \( a^{2} - 1\) , \( -15 a^{3} - 85 a^{2} + 385 a - 331\) , \( 187 a^{3} - 1626 a^{2} + 3723 a - 2518\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-3a\right){x}^{2}+\left(-15a^{3}-85a^{2}+385a-331\right){x}+187a^{3}-1626a^{2}+3723a-2518$
31.2-a2 31.2-a \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/4\Z$ $1$ $226.2402485$ 0.843147624 \( -\frac{29296819153015039230435114609}{923521} a^{3} - \frac{9909977575470350951831216400}{923521} a^{2} + \frac{103925138003905943940922438084}{923521} a + \frac{21891704604902707693397138361}{923521} \) \( \bigl[a + 1\) , \( a^{3} - 3 a\) , \( a^{2} - 1\) , \( 270 a^{3} - 70 a^{2} - 655 a - 231\) , \( -2943 a^{3} + 1542 a^{2} + 6186 a + 1507\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-3a\right){x}^{2}+\left(270a^{3}-70a^{2}-655a-231\right){x}-2943a^{3}+1542a^{2}+6186a+1507$
31.2-a3 31.2-a \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $226.2402485$ 0.843147624 \( -\frac{96230619458397643875589}{852891037441} a^{3} - \frac{32551084210297485030913}{852891037441} a^{2} + \frac{341360629620385537992211}{852891037441} a + \frac{71907211552629246852068}{852891037441} \) \( \bigl[a + 1\) , \( a^{3} - 3 a\) , \( a^{2} - 1\) , \( 15 a^{3} - 5 a^{2} - 25 a - 31\) , \( -47 a^{3} + 4 a^{2} + 161 a - 22\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-3a\right){x}^{2}+\left(15a^{3}-5a^{2}-25a-31\right){x}-47a^{3}+4a^{2}+161a-22$
31.2-a4 31.2-a \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $226.2402485$ 0.843147624 \( \frac{12067174606324}{923521} a^{3} - \frac{14868316428812}{923521} a^{2} - \frac{45335200487011}{923521} a + \frac{58867032461318}{923521} \) \( \bigl[a + 1\) , \( a^{3} - 3 a\) , \( a^{2} - 1\) , \( -5 a^{2} + 20 a - 21\) , \( -21 a^{2} + 62 a - 44\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-3a\right){x}^{2}+\left(-5a^{2}+20a-21\right){x}-21a^{2}+62a-44$
31.2-a5 31.2-a \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/4\Z$ $1$ $226.2402485$ 0.843147624 \( -\frac{1015616}{961} a^{3} + \frac{1884681}{961} a^{2} + \frac{4181756}{961} a - \frac{6745220}{961} \) \( \bigl[a + 1\) , \( a^{3} - 3 a\) , \( a^{2} - 1\) , \( -1\) , \( -a^{2} + 2 a - 1\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-3a\right){x}^{2}-{x}-a^{2}+2a-1$
31.2-a6 31.2-a \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/2\Z$ $1$ $14.14001553$ 0.843147624 \( \frac{154534453074689854099149438193}{727423121747185263828481} a^{3} + \frac{52426907107682729263745202448}{727423121747185263828481} a^{2} - \frac{548494138857606744358415989700}{727423121747185263828481} a - \frac{115417448688070173417441163369}{727423121747185263828481} \) \( \bigl[a + 1\) , \( a^{3} - 3 a\) , \( a^{2} - 1\) , \( 60 a^{2} - 115 a + 9\) , \( -59 a^{3} - 14 a^{2} + 312 a - 163\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-3a\right){x}^{2}+\left(60a^{2}-115a+9\right){x}-59a^{3}-14a^{2}+312a-163$
31.2-b1 31.2-b \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/2\Z$ $1$ $2.959428063$ 0.705864781 \( -\frac{29296819153015039230435114609}{923521} a^{3} - \frac{9909977575470350951831216400}{923521} a^{2} + \frac{103925138003905943940922438084}{923521} a + \frac{21891704604902707693397138361}{923521} \) \( \bigl[a^{3} - 2 a + 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( a^{2} - 1\) , \( 416 a^{3} + 76 a^{2} - 1585 a - 348\) , \( 6396 a^{3} + 1587 a^{2} - 23865 a - 5050\bigr] \) ${y}^2+\left(a^{3}-2a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a+2\right){x}^{2}+\left(416a^{3}+76a^{2}-1585a-348\right){x}+6396a^{3}+1587a^{2}-23865a-5050$
31.2-b2 31.2-b \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/8\Z$ $1$ $757.6135842$ 0.705864781 \( \frac{934917206863109317}{961} a^{3} - \frac{1130359602108196159}{961} a^{2} - \frac{3503384511067738195}{961} a + \frac{4472090816549444332}{961} \) \( \bigl[a^{3} - 2 a + 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( a^{2} - 1\) , \( -24 a^{3} - 4 a^{2} + 90 a - 63\) , \( 49 a^{3} + 26 a^{2} - 250 a + 169\bigr] \) ${y}^2+\left(a^{3}-2a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a+2\right){x}^{2}+\left(-24a^{3}-4a^{2}+90a-63\right){x}+49a^{3}+26a^{2}-250a+169$
31.2-b3 31.2-b \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $47.35084901$ 0.705864781 \( -\frac{96230619458397643875589}{852891037441} a^{3} - \frac{32551084210297485030913}{852891037441} a^{2} + \frac{341360629620385537992211}{852891037441} a + \frac{71907211552629246852068}{852891037441} \) \( \bigl[a^{3} - 2 a + 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( a^{2} - 1\) , \( 26 a^{3} + 6 a^{2} - 100 a - 23\) , \( 93 a^{3} + 20 a^{2} - 358 a - 75\bigr] \) ${y}^2+\left(a^{3}-2a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a+2\right){x}^{2}+\left(26a^{3}+6a^{2}-100a-23\right){x}+93a^{3}+20a^{2}-358a-75$
31.2-b4 31.2-b \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/2\Z$ $1$ $2.959428063$ 0.705864781 \( \frac{154534453074689854099149438193}{727423121747185263828481} a^{3} + \frac{52426907107682729263745202448}{727423121747185263828481} a^{2} - \frac{548494138857606744358415989700}{727423121747185263828481} a - \frac{115417448688070173417441163369}{727423121747185263828481} \) \( \bigl[a^{3} - 2 a + 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( a^{2} - 1\) , \( 36 a^{3} + 16 a^{2} - 135 a - 18\) , \( 78 a^{3} - 11 a^{2} - 319 a - 36\bigr] \) ${y}^2+\left(a^{3}-2a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a+2\right){x}^{2}+\left(36a^{3}+16a^{2}-135a-18\right){x}+78a^{3}-11a^{2}-319a-36$
31.2-b5 31.2-b \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/2\Z\oplus\Z/4\Z$ $1$ $757.6135842$ 0.705864781 \( \frac{12067174606324}{923521} a^{3} - \frac{14868316428812}{923521} a^{2} - \frac{45335200487011}{923521} a + \frac{58867032461318}{923521} \) \( \bigl[a^{3} - 2 a + 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( a^{2} - 1\) , \( a^{3} + a^{2} - 5 a - 3\) , \( a^{3} + a^{2} - 6 a - 1\bigr] \) ${y}^2+\left(a^{3}-2a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a+2\right){x}^{2}+\left(a^{3}+a^{2}-5a-3\right){x}+a^{3}+a^{2}-6a-1$
31.2-b6 31.2-b \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/8\Z$ $1$ $757.6135842$ 0.705864781 \( -\frac{1015616}{961} a^{3} + \frac{1884681}{961} a^{2} + \frac{4181756}{961} a - \frac{6745220}{961} \) \( \bigl[a^{3} - 2 a + 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( a^{2} - 1\) , \( a^{3} + a^{2} - 5 a + 2\) , \( a^{3} - 2 a\bigr] \) ${y}^2+\left(a^{3}-2a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a+2\right){x}^{2}+\left(a^{3}+a^{2}-5a+2\right){x}+a^{3}-2a$
31.3-a1 31.3-a \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/4\Z$ $1$ $226.2402485$ 0.843147624 \( -\frac{3352161032683862028986803952}{923521} a^{3} + \frac{9909977575470350951831216400}{923521} a^{2} - \frac{5978197446809240162656682401}{923521} a - \frac{1713525152117869864310632982}{923521} \) \( \bigl[a^{3} - 2 a\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( 495 a^{3} + 69 a^{2} - 1642 a - 353\) , \( -7129 a^{3} - 1543 a^{2} + 24030 a + 5034\bigr] \) ${y}^2+\left(a^{3}-2a\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+2\right){x}^{2}+\left(495a^{3}+69a^{2}-1642a-353\right){x}-7129a^{3}-1543a^{2}+24030a+5034$
31.3-a2 31.3-a \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/2\Z$ $1$ $14.14001553$ 0.843147624 \( \frac{1366643918492895232}{961} a^{3} + \frac{1130359602108196159}{961} a^{2} - \frac{3401298865000275452}{961} a - \frac{747980482361750548}{961} \) \( \bigl[a^{3} - 2 a\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( 410 a^{3} + 84 a^{2} - 1572 a - 328\) , \( 6096 a^{3} + 1625 a^{2} - 22572 a - 4736\bigr] \) ${y}^2+\left(a^{3}-2a\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+2\right){x}^{2}+\left(410a^{3}+84a^{2}-1572a-328\right){x}+6096a^{3}+1625a^{2}-22572a-4736$
31.3-a3 31.3-a \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $226.2402485$ 0.843147624 \( -\frac{11010764002907552479232}{852891037441} a^{3} + \frac{32551084210297485030913}{852891037441} a^{2} - \frac{19636479236469948927748}{852891037441} a - \frac{5628354043368086906140}{852891037441} \) \( \bigl[a^{3} - 2 a\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( 40 a^{3} + 4 a^{2} - 142 a - 28\) , \( -32 a^{3} - 5 a^{2} + 76 a + 16\bigr] \) ${y}^2+\left(a^{3}-2a\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+2\right){x}^{2}+\left(40a^{3}+4a^{2}-142a-28\right){x}-32a^{3}-5a^{2}+76a+16$
31.3-a4 31.3-a \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $226.2402485$ 0.843147624 \( \frac{17801814367097}{923521} a^{3} + \frac{14868316428812}{923521} a^{2} - \frac{44271766433252}{923521} a - \frac{9739909921969}{923521} \) \( \bigl[a^{3} - 2 a\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( 25 a^{3} + 4 a^{2} - 97 a - 18\) , \( 82 a^{3} + 20 a^{2} - 308 a - 64\bigr] \) ${y}^2+\left(a^{3}-2a\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+2\right){x}^{2}+\left(25a^{3}+4a^{2}-97a-18\right){x}+82a^{3}+20a^{2}-308a-64$
31.3-a5 31.3-a \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/4\Z$ $1$ $226.2402485$ 0.843147624 \( -\frac{1765389}{961} a^{3} - \frac{1884681}{961} a^{2} + \frac{4161259}{961} a + \frac{1928412}{961} \) \( \bigl[a^{3} - 2 a\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -a^{2} - 2 a + 2\) , \( 2 a^{3} - 8 a - 1\bigr] \) ${y}^2+\left(a^{3}-2a\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+2\right){x}^{2}+\left(-a^{2}-2a+2\right){x}+2a^{3}-8a-1$
31.3-a6 31.3-a \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/2\Z$ $1$ $14.14001553$ 0.843147624 \( \frac{17216766333469942774436560624}{727423121747185263828481} a^{3} - \frac{52426907107682729263745202448}{727423121747185263828481} a^{2} + \frac{33240480633127353737657993249}{727423121747185263828481} a + \frac{9399400109123561576571971302}{727423121747185263828481} \) \( \bigl[a^{3} - 2 a\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -175 a^{3} - 61 a^{2} + 638 a + 137\) , \( 89 a^{3} + 13 a^{2} - 402 a - 82\bigr] \) ${y}^2+\left(a^{3}-2a\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+2\right){x}^{2}+\left(-175a^{3}-61a^{2}+638a+137\right){x}+89a^{3}+13a^{2}-402a-82$
31.3-b1 31.3-b \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/2\Z$ $1$ $2.959428063$ 0.705864781 \( -\frac{3352161032683862028986803952}{923521} a^{3} + \frac{9909977575470350951831216400}{923521} a^{2} - \frac{5978197446809240162656682401}{923521} a - \frac{1713525152117869864310632982}{923521} \) \( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 5 a^{3} - 75 a^{2} + 320 a - 384\) , \( 330 a^{3} - 1847 a^{2} + 3763 a - 2740\bigr] \) ${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(5a^{3}-75a^{2}+320a-384\right){x}+330a^{3}-1847a^{2}+3763a-2740$
31.3-b2 31.3-b \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/8\Z$ $1$ $757.6135842$ 0.705864781 \( \frac{1366643918492895232}{961} a^{3} + \frac{1130359602108196159}{961} a^{2} - \frac{3401298865000275452}{961} a - \frac{747980482361750548}{961} \) \( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 5 a^{2} - 20 a - 64\) , \( -7 a^{3} - 11 a^{2} + 120 a + 219\bigr] \) ${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(5a^{2}-20a-64\right){x}-7a^{3}-11a^{2}+120a+219$
31.3-b3 31.3-b \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $47.35084901$ 0.705864781 \( -\frac{11010764002907552479232}{852891037441} a^{3} + \frac{32551084210297485030913}{852891037441} a^{2} - \frac{19636479236469948927748}{852891037441} a - \frac{5628354043368086906140}{852891037441} \) \( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -5 a^{2} + 20 a - 24\) , \( 7 a^{3} - 35 a^{2} + 64 a - 35\bigr] \) ${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-5a^{2}+20a-24\right){x}+7a^{3}-35a^{2}+64a-35$
31.3-b4 31.3-b \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/2\Z\oplus\Z/4\Z$ $1$ $757.6135842$ 0.705864781 \( \frac{17801814367097}{923521} a^{3} + \frac{14868316428812}{923521} a^{2} - \frac{44271766433252}{923521} a - \frac{9739909921969}{923521} \) \( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -4\) , \( -a^{2} + 4 a + 4\bigr] \) ${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}-4{x}-a^{2}+4a+4$
31.3-b5 31.3-b \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/8\Z$ $1$ $757.6135842$ 0.705864781 \( -\frac{1765389}{961} a^{3} - \frac{1884681}{961} a^{2} + \frac{4161259}{961} a + \frac{1928412}{961} \) \( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 1\) , \( 0\bigr] \) ${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+{x}$
31.3-b6 31.3-b \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/2\Z$ $1$ $2.959428063$ 0.705864781 \( \frac{17216766333469942774436560624}{727423121747185263828481} a^{3} - \frac{52426907107682729263745202448}{727423121747185263828481} a^{2} + \frac{33240480633127353737657993249}{727423121747185263828481} a + \frac{9399400109123561576571971302}{727423121747185263828481} \) \( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -5 a^{3} - 15 a^{2} + 40 a + 16\) , \( -8 a^{3} + a^{2} + 125 a - 166\bigr] \) ${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-5a^{3}-15a^{2}+40a+16\right){x}-8a^{3}+a^{2}+125a-166$
31.4-a1 31.4-a \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/2\Z$ $1$ $14.14001553$ 0.843147624 \( -\frac{2065276808971305476}{961} a^{3} - \frac{698632890478410244}{961} a^{2} + \frac{7326190029022112587}{961} a + \frac{1543360584318567026}{961} \) \( \bigl[a^{2} - 1\) , \( -a^{3} + 3 a - 1\) , \( a^{3} - 2 a\) , \( -71 a^{3} + 341 a^{2} + 296 a - 1252\) , \( -1813 a^{3} + 4283 a^{2} + 7065 a - 16149\bigr] \) ${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}-2a\right){y}={x}^{3}+\left(-a^{3}+3a-1\right){x}^{2}+\left(-71a^{3}+341a^{2}+296a-1252\right){x}-1813a^{3}+4283a^{2}+7065a-16149$
31.4-a2 31.4-a \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/4\Z$ $1$ $226.2402485$ 0.843147624 \( \frac{19386841577544688278603898209}{923521} a^{3} + \frac{16034680544860826249617094257}{923521} a^{2} - \frac{48250547157163713883980478227}{923521} a - \frac{10610770058214958430895584744}{923521} \) \( \bigl[a^{2} - 1\) , \( -a^{3} + 3 a - 1\) , \( a^{3} - 2 a\) , \( -341 a^{3} + 156 a^{2} + 1091 a - 1022\) , \( 4485 a^{3} - 2644 a^{2} - 14997 a + 14364\bigr] \) ${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}-2a\right){y}={x}^{3}+\left(-a^{3}+3a-1\right){x}^{2}+\left(-341a^{3}+156a^{2}+1091a-1022\right){x}+4485a^{3}-2644a^{2}-14997a+14364$
31.4-a3 31.4-a \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $226.2402485$ 0.843147624 \( \frac{63679535248100158844676}{852891037441} a^{3} + \frac{52668771245192606365444}{852891037441} a^{2} - \frac{158487521534002991503115}{852891037441} a - \frac{34852964110250777095970}{852891037441} \) \( \bigl[a^{2} - 1\) , \( -a^{3} + 3 a - 1\) , \( a^{3} - 2 a\) , \( -21 a^{3} + 21 a^{2} + 66 a - 102\) , \( 51 a^{3} + 19 a^{2} - 157 a - 1\bigr] \) ${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}-2a\right){y}={x}^{3}+\left(-a^{3}+3a-1\right){x}^{2}+\left(-21a^{3}+21a^{2}+66a-102\right){x}+51a^{3}+19a^{2}-157a-1$
31.4-a4 31.4-a \(\Q(\zeta_{15})^+\) \( 31 \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $226.2402485$ 0.843147624 \( -\frac{26935491035136}{923521} a^{3} - \frac{9133676668039}{923521} a^{2} + \frac{95674789534220}{923521} a + \frac{20462261904636}{923521} \) \( \bigl[a^{2} - 1\) , \( -a^{3} + 3 a - 1\) , \( a^{3} - 2 a\) , \( -6 a^{3} + 21 a^{2} + 21 a - 77\) , \( -21 a^{3} + 61 a^{2} + 84 a - 229\bigr] \) ${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}-2a\right){y}={x}^{3}+\left(-a^{3}+3a-1\right){x}^{2}+\left(-6a^{3}+21a^{2}+21a-77\right){x}-21a^{3}+61a^{2}+84a-229$
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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.