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Results (28 matches)

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Label Class Base field Conductor norm Rank Torsion CM Sato-Tate Regulator Period Leading coeff j-invariant Weierstrass coefficients Weierstrass equation
25.2-a1 25.2-a \(\Q(\sqrt{21}) \) \( 5^{2} \) 0 $\Z/2\Z$ $-7$ $N(\mathrm{U}(1))$ $1$ $11.70083348$ 1.276665598 \( -3375 \) \( \bigl[a\) , \( -a + 1\) , \( a + 1\) , \( -a - 3\) , \( -a - 1\bigr] \) ${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-a-3\right){x}-a-1$
25.2-a2 25.2-a \(\Q(\sqrt{21}) \) \( 5^{2} \) 0 $\Z/2\Z$ $-7$ $N(\mathrm{U}(1))$ $1$ $11.70083348$ 1.276665598 \( -3375 \) \( \bigl[1\) , \( -1\) , \( a + 1\) , \( -3 a - 5\) , \( 4 a + 7\bigr] \) ${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-3a-5\right){x}+4a+7$
25.3-a1 25.3-a \(\Q(\sqrt{21}) \) \( 5^{2} \) 0 $\Z/2\Z$ $-7$ $N(\mathrm{U}(1))$ $1$ $11.70083348$ 1.276665598 \( -3375 \) \( \bigl[1\) , \( -1\) , \( a\) , \( 2 a - 7\) , \( -5 a + 12\bigr] \) ${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(2a-7\right){x}-5a+12$
25.3-a2 25.3-a \(\Q(\sqrt{21}) \) \( 5^{2} \) 0 $\Z/2\Z$ $-7$ $N(\mathrm{U}(1))$ $1$ $11.70083348$ 1.276665598 \( -3375 \) \( \bigl[a + 1\) , \( 0\) , \( a\) , \( -a - 3\) , \( -1\bigr] \) ${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-3\right){x}-1$
49.1-b1 49.1-b \(\Q(\sqrt{21}) \) \( 7^{2} \) $1$ $\Z/2\Z$ $-7$ $N(\mathrm{U}(1))$ $0.109590133$ $26.16385905$ 1.251392664 \( -3375 \) \( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 10 a - 28\) , \( -24 a + 67\bigr] \) ${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(10a-28\right){x}-24a+67$
49.1-b2 49.1-b \(\Q(\sqrt{21}) \) \( 7^{2} \) $1$ $\Z/2\Z$ $-7$ $N(\mathrm{U}(1))$ $0.767130934$ $3.737694151$ 1.251392664 \( -3375 \) \( \bigl[1\) , \( -1\) , \( 0\) , \( -2\) , \( -1\bigr] \) ${y}^2+{x}{y}={x}^{3}-{x}^{2}-2{x}-1$
225.2-d1 225.2-d \(\Q(\sqrt{21}) \) \( 3^{2} \cdot 5^{2} \) $1$ $\Z/2\Z$ $-7$ $N(\mathrm{U}(1))$ $0.247061726$ $17.87331837$ 1.927218749 \( -3375 \) \( \bigl[1\) , \( -1\) , \( 0\) , \( 15 a - 42\) , \( -63 a + 176\bigr] \) ${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(15a-42\right){x}-63a+176$
225.2-d2 225.2-d \(\Q(\sqrt{21}) \) \( 3^{2} \cdot 5^{2} \) $1$ $\Z/2\Z$ $-7$ $N(\mathrm{U}(1))$ $1.729432086$ $2.553331196$ 1.927218749 \( -3375 \) \( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -3\) , \( -3 a - 8\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}-3{x}-3a-8$
225.3-d1 225.3-d \(\Q(\sqrt{21}) \) \( 3^{2} \cdot 5^{2} \) $1$ $\Z/2\Z$ $-7$ $N(\mathrm{U}(1))$ $1.729432086$ $2.553331196$ 1.927218749 \( -3375 \) \( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -3\) , \( 3 a - 11\bigr] \) ${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}-3{x}+3a-11$
225.3-d2 225.3-d \(\Q(\sqrt{21}) \) \( 3^{2} \cdot 5^{2} \) $1$ $\Z/2\Z$ $-7$ $N(\mathrm{U}(1))$ $0.247061726$ $17.87331837$ 1.927218749 \( -3375 \) \( \bigl[1\) , \( -1\) , \( 0\) , \( -15 a - 27\) , \( 63 a + 113\bigr] \) ${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(-15a-27\right){x}+63a+113$
256.1-b1 256.1-b \(\Q(\sqrt{21}) \) \( 2^{8} \) $1$ $\Z/2\Z$ $-7$ $N(\mathrm{U}(1))$ $2.329446770$ $2.472252300$ 2.513424990 \( -3375 \) \( \bigl[0\) , \( 0\) , \( 0\) , \( 5 a - 15\) , \( 12 a - 34\bigr] \) ${y}^2={x}^{3}+\left(5a-15\right){x}+12a-34$
256.1-b2 256.1-b \(\Q(\sqrt{21}) \) \( 2^{8} \) $1$ $\Z/2\Z$ $-7$ $N(\mathrm{U}(1))$ $0.332778110$ $17.30576610$ 2.513424990 \( -3375 \) \( \bigl[0\) , \( 0\) , \( 0\) , \( -5 a - 10\) , \( 12 a + 22\bigr] \) ${y}^2={x}^{3}+\left(-5a-10\right){x}+12a+22$
256.1-j1 256.1-j \(\Q(\sqrt{21}) \) \( 2^{8} \) $1$ $\Z/2\Z$ $-7$ $N(\mathrm{U}(1))$ $0.332778110$ $17.30576610$ 2.513424990 \( -3375 \) \( \bigl[0\) , \( 0\) , \( 0\) , \( 5 a - 15\) , \( -12 a + 34\bigr] \) ${y}^2={x}^{3}+\left(5a-15\right){x}-12a+34$
256.1-j2 256.1-j \(\Q(\sqrt{21}) \) \( 2^{8} \) $1$ $\Z/2\Z$ $-7$ $N(\mathrm{U}(1))$ $2.329446770$ $2.472252300$ 2.513424990 \( -3375 \) \( \bigl[0\) , \( 0\) , \( 0\) , \( -5 a - 10\) , \( -12 a - 22\bigr] \) ${y}^2={x}^{3}+\left(-5a-10\right){x}-12a-22$
289.2-d1 289.2-d \(\Q(\sqrt{21}) \) \( 17^{2} \) 0 $\Z/2\Z$ $-7$ $N(\mathrm{U}(1))$ $1$ $6.345667909$ 0.692369131 \( -3375 \) \( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -15 a - 23\) , \( 53 a + 96\bigr] \) ${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-15a-23\right){x}+53a+96$
289.2-d2 289.2-d \(\Q(\sqrt{21}) \) \( 17^{2} \) 0 $\Z/2\Z$ $-7$ $N(\mathrm{U}(1))$ $1$ $6.345667909$ 0.692369131 \( -3375 \) \( \bigl[a + 1\) , \( 0\) , \( 1\) , \( 2 a - 6\) , \( -4 a + 7\bigr] \) ${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(2a-6\right){x}-4a+7$
289.3-d1 289.3-d \(\Q(\sqrt{21}) \) \( 17^{2} \) 0 $\Z/2\Z$ $-7$ $N(\mathrm{U}(1))$ $1$ $6.345667909$ 0.692369131 \( -3375 \) \( \bigl[a\) , \( -a + 1\) , \( 1\) , \( -3 a - 3\) , \( 4 a + 3\bigr] \) ${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-3a-3\right){x}+4a+3$
289.3-d2 289.3-d \(\Q(\sqrt{21}) \) \( 17^{2} \) 0 $\Z/2\Z$ $-7$ $N(\mathrm{U}(1))$ $1$ $6.345667909$ 0.692369131 \( -3375 \) \( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 15 a - 38\) , \( -53 a + 149\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(15a-38\right){x}-53a+149$
441.1-c1 441.1-c \(\Q(\sqrt{21}) \) \( 3^{2} \cdot 7^{2} \) 0 $\Z/2\Z$ $-7$ $N(\mathrm{U}(1))$ $1$ $5.709422123$ 2.491796100 \( -3375 \) \( \bigl[a\) , \( -a + 1\) , \( a\) , \( -8 a - 13\) , \( -19 a - 34\bigr] \) ${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-8a-13\right){x}-19a-34$
441.1-c2 441.1-c \(\Q(\sqrt{21}) \) \( 3^{2} \cdot 7^{2} \) 0 $\Z/2\Z$ $-7$ $N(\mathrm{U}(1))$ $1$ $5.709422123$ 2.491796100 \( -3375 \) \( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 6 a - 21\) , \( 18 a - 53\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(6a-21\right){x}+18a-53$
1681.2-b1 1681.2-b \(\Q(\sqrt{21}) \) \( 41^{2} \) 0 $\Z/2\Z$ $-7$ $N(\mathrm{U}(1))$ $1$ $4.086108294$ 0.445830965 \( -3375 \) \( \bigl[a\) , \( -a + 1\) , \( 1\) , \( 27 a - 78\) , \( 190 a - 530\bigr] \) ${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(27a-78\right){x}+190a-530$
1681.2-b2 1681.2-b \(\Q(\sqrt{21}) \) \( 41^{2} \) 0 $\Z/2\Z$ $-7$ $N(\mathrm{U}(1))$ $1$ $4.086108294$ 0.445830965 \( -3375 \) \( \bigl[1\) , \( -1\) , \( 1\) , \( -5 a - 15\) , \( -16 a - 21\bigr] \) ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-5a-15\right){x}-16a-21$
1681.3-b1 1681.3-b \(\Q(\sqrt{21}) \) \( 41^{2} \) 0 $\Z/2\Z$ $-7$ $N(\mathrm{U}(1))$ $1$ $4.086108294$ 0.445830965 \( -3375 \) \( \bigl[1\) , \( -1\) , \( 1\) , \( 5 a - 20\) , \( 16 a - 37\bigr] \) ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(5a-20\right){x}+16a-37$
1681.3-b2 1681.3-b \(\Q(\sqrt{21}) \) \( 41^{2} \) 0 $\Z/2\Z$ $-7$ $N(\mathrm{U}(1))$ $1$ $4.086108294$ 0.445830965 \( -3375 \) \( \bigl[a + 1\) , \( 0\) , \( 1\) , \( -28 a - 51\) , \( -190 a - 340\bigr] \) ${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-28a-51\right){x}-190a-340$
1849.2-a1 1849.2-a \(\Q(\sqrt{21}) \) \( 43^{2} \) $1$ $\Z/2\Z$ $-7$ $N(\mathrm{U}(1))$ $1.179276195$ $10.55641835$ 5.433159735 \( -3375 \) \( \bigl[a\) , \( -a + 1\) , \( a\) , \( -3 a - 13\) , \( 6 a + 15\bigr] \) ${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-3a-13\right){x}+6a+15$
1849.2-a2 1849.2-a \(\Q(\sqrt{21}) \) \( 43^{2} \) $1$ $\Z/2\Z$ $-7$ $N(\mathrm{U}(1))$ $8.254933368$ $1.508059765$ 5.433159735 \( -3375 \) \( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 51 a - 146\) , \( 379 a - 1061\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(51a-146\right){x}+379a-1061$
1849.3-a1 1849.3-a \(\Q(\sqrt{21}) \) \( 43^{2} \) $1$ $\Z/2\Z$ $-7$ $N(\mathrm{U}(1))$ $8.254933368$ $1.508059765$ 5.433159735 \( -3375 \) \( \bigl[a\) , \( -a + 1\) , \( a\) , \( -53 a - 93\) , \( -380 a - 681\bigr] \) ${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-53a-93\right){x}-380a-681$
1849.3-a2 1849.3-a \(\Q(\sqrt{21}) \) \( 43^{2} \) $1$ $\Z/2\Z$ $-7$ $N(\mathrm{U}(1))$ $1.179276195$ $10.55641835$ 5.433159735 \( -3375 \) \( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( a - 16\) , \( -7 a + 21\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-16\right){x}-7a+21$
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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.