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Base field Conductor norm Rank* Torsion order CM discriminant
Base field degree/signature Bad \(p\) $\Q$-curves Torsion structure CM
Analytic order of Ш* Cyclic isogeny degree Isogeny class size Reduction Galois image
j-invariant Regulator* Curves per isogeny class Isogeny class degree Nonmax$\ \ell$
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Results (6 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
8.2-a1 8.2-a 4.4.2777.1 \( 2^{3} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $91.38443470$ 0.867070267 \( 8986661761218306693 a^{3} + 12242887281724452580 a^{2} - 7024786750071806011 a - 7608271795805318134 \) \( \bigl[a^{2} - a - 2\) , \( -a^{3} + a^{2} + 2 a - 2\) , \( a^{3} - 4 a\) , \( -2 a^{3} - 9 a^{2} - 10 a + 2\) , \( 32 a^{3} + 44 a^{2} - 34 a - 39\bigr] \) ${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+2a-2\right){x}^{2}+\left(-2a^{3}-9a^{2}-10a+2\right){x}+32a^{3}+44a^{2}-34a-39$
8.2-a2 8.2-a 4.4.2777.1 \( 2^{3} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $182.7688694$ 0.867070267 \( -601 a^{3} + 1330 a^{2} + 1931 a - 2180 \) \( \bigl[a^{2} - a - 2\) , \( -a^{3} + a^{2} + 4 a\) , \( a^{3} - a^{2} - 2 a + 2\) , \( -a^{3} - a^{2} + 4 a + 5\) , \( -a^{3} - a^{2} + 4 a + 1\bigr] \) ${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-a^{2}-2a+2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a\right){x}^{2}+\left(-a^{3}-a^{2}+4a+5\right){x}-a^{3}-a^{2}+4a+1$
8.2-a3 8.2-a 4.4.2777.1 \( 2^{3} \) 0 $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $365.5377388$ 0.867070267 \( -2732527394 a^{3} + 1558959970 a^{2} + 13692770763 a + 7728253510 \) \( \bigl[a^{2} - 2\) , \( a^{3} - 3 a - 2\) , \( 0\) , \( 9 a^{3} - 25 a^{2} + 14 a + 4\) , \( -30 a^{3} + 72 a^{2} + 25 a - 54\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(a^{3}-3a-2\right){x}^{2}+\left(9a^{3}-25a^{2}+14a+4\right){x}-30a^{3}+72a^{2}+25a-54$
8.2-a4 8.2-a 4.4.2777.1 \( 2^{3} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $91.38443470$ 0.867070267 \( -63795770348435003481 a^{3} + 11114208719071878840 a^{2} + 264361023260378305267 a + 154509478780590380994 \) \( \bigl[a^{2} - 2\) , \( a^{3} - 3 a - 2\) , \( 0\) , \( 154 a^{3} - 405 a^{2} + 39 a + 154\) , \( -2510 a^{3} + 6386 a^{2} + 264 a - 3160\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(a^{3}-3a-2\right){x}^{2}+\left(154a^{3}-405a^{2}+39a+154\right){x}-2510a^{3}+6386a^{2}+264a-3160$
8.2-a5 8.2-a 4.4.2777.1 \( 2^{3} \) 0 $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $365.5377388$ 0.867070267 \( 2973144 a^{3} - 5002655 a^{2} - 8485568 a + 8881408 \) \( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a + 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( 3 a^{3} - 2 a^{2} - 9 a - 6\) , \( 6 a^{3} - 3 a^{2} - 22 a - 13\bigr] \) ${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a^{3}-a^{2}-2a+2\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(3a^{3}-2a^{2}-9a-6\right){x}+6a^{3}-3a^{2}-22a-13$
8.2-a6 8.2-a 4.4.2777.1 \( 2^{3} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $45.69221735$ 0.867070267 \( 34259927531436 a^{3} - 57543520518160 a^{2} - 97932989693671 a + 100819215025068 \) \( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a + 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( 18 a^{3} - 37 a^{2} - 14 a + 4\) , \( 63 a^{3} - 160 a^{2} - 5 a + 73\bigr] \) ${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a^{3}-a^{2}-2a+2\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(18a^{3}-37a^{2}-14a+4\right){x}+63a^{3}-160a^{2}-5a+73$
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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.