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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 (24 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
729.1-a1 729.1-a \(\Q(\sqrt{13}) \) \( 3^{6} \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $6.546120139$ 1.815567063 \( -13515 a + 30222 \) \( \bigl[a\) , \( -a\) , \( a + 1\) , \( -13 a - 21\) , \( 65 a + 78\bigr] \) ${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-13a-21\right){x}+65a+78$
729.1-a2 729.1-a \(\Q(\sqrt{13}) \) \( 3^{6} \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $6.546120139$ 1.815567063 \( -16755 a + 39237 \) \( \bigl[a + 1\) , \( -1\) , \( a\) , \( 38 a + 48\) , \( -201 a - 261\bigr] \) ${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-{x}^{2}+\left(38a+48\right){x}-201a-261$
729.1-b1 729.1-b \(\Q(\sqrt{13}) \) \( 3^{6} \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $4.860164952$ 1.347967226 \( -13515 a + 30222 \) \( \bigl[a + 1\) , \( -1\) , \( 1\) , \( -2 a - 5\) , \( -6 a - 9\bigr] \) ${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-{x}^{2}+\left(-2a-5\right){x}-6a-9$
729.1-b2 729.1-b \(\Q(\sqrt{13}) \) \( 3^{6} \) 0 $\Z/3\Z$ $\mathrm{SU}(2)$ $1$ $14.58049485$ 1.347967226 \( -16755 a + 39237 \) \( \bigl[1\) , \( -1\) , \( a\) , \( 7 a + 9\) , \( 16 a + 21\bigr] \) ${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(7a+9\right){x}+16a+21$
729.1-c1 729.1-c \(\Q(\sqrt{13}) \) \( 3^{6} \) 0 $\mathsf{trivial}$ $-27$ $N(\mathrm{U}(1))$ $1$ $3.121012474$ 2.596839347 \( -12288000 \) \( \bigl[0\) , \( 0\) , \( 1\) , \( 30 a - 120\) , \( 253 a - 443\bigr] \) ${y}^2+{y}={x}^{3}+\left(30a-120\right){x}+253a-443$
729.1-c2 729.1-c \(\Q(\sqrt{13}) \) \( 3^{6} \) 0 $\mathsf{trivial}$ $-27$ $N(\mathrm{U}(1))$ $1$ $3.121012474$ 2.596839347 \( -12288000 \) \( \bigl[0\) , \( 0\) , \( 1\) , \( -30 a - 90\) , \( -253 a - 190\bigr] \) ${y}^2+{y}={x}^{3}+\left(-30a-90\right){x}-253a-190$
729.1-c3 729.1-c \(\Q(\sqrt{13}) \) \( 3^{6} \) 0 $\mathsf{trivial}$ $-3$ $N(\mathrm{U}(1))$ $1$ $9.363037422$ 2.596839347 \( 0 \) \( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( a - 2\bigr] \) ${y}^2+{y}={x}^{3}+a-2$
729.1-c4 729.1-c \(\Q(\sqrt{13}) \) \( 3^{6} \) 0 $\mathsf{trivial}$ $-3$ $N(\mathrm{U}(1))$ $1$ $9.363037422$ 2.596839347 \( 0 \) \( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( -a - 1\bigr] \) ${y}^2+{y}={x}^{3}-a-1$
729.1-d1 729.1-d \(\Q(\sqrt{13}) \) \( 3^{6} \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.354151165$ $14.41546786$ 2.831885807 \( 13515 a + 16707 \) \( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -5\) , \( a - 5\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}-5{x}+a-5$
729.1-d2 729.1-d \(\Q(\sqrt{13}) \) \( 3^{6} \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.118050388$ $14.41546786$ 2.831885807 \( 16755 a + 22482 \) \( \bigl[a\) , \( -a\) , \( a\) , \( -5 a + 9\) , \( -11 a + 24\bigr] \) ${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-5a+9\right){x}-11a+24$
729.1-e1 729.1-e \(\Q(\sqrt{13}) \) \( 3^{6} \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $4.860164952$ 1.347967226 \( 13515 a + 16707 \) \( \bigl[a\) , \( -a\) , \( 1\) , \( a - 6\) , \( 6 a - 15\bigr] \) ${y}^2+a{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(a-6\right){x}+6a-15$
729.1-e2 729.1-e \(\Q(\sqrt{13}) \) \( 3^{6} \) 0 $\Z/3\Z$ $\mathrm{SU}(2)$ $1$ $14.58049485$ 1.347967226 \( 16755 a + 22482 \) \( \bigl[1\) , \( -1\) , \( a + 1\) , \( -8 a + 16\) , \( -17 a + 37\bigr] \) ${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-8a+16\right){x}-17a+37$
729.1-f1 729.1-f \(\Q(\sqrt{13}) \) \( 3^{6} \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $6.546120139$ 1.815567063 \( 13515 a + 16707 \) \( \bigl[a + 1\) , \( -1\) , \( a\) , \( 11 a - 33\) , \( -66 a + 144\bigr] \) ${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-{x}^{2}+\left(11a-33\right){x}-66a+144$
729.1-f2 729.1-f \(\Q(\sqrt{13}) \) \( 3^{6} \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $6.546120139$ 1.815567063 \( 16755 a + 22482 \) \( \bigl[a\) , \( -a\) , \( a + 1\) , \( -40 a + 87\) , \( 200 a - 462\bigr] \) ${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-40a+87\right){x}+200a-462$
729.1-g1 729.1-g \(\Q(\sqrt{13}) \) \( 3^{6} \) $1$ $\Z/3\Z$ $\mathrm{SU}(2)$ $0.227641124$ $19.41608679$ 2.451719308 \( 13515 a + 16707 \) \( \bigl[1\) , \( -1\) , \( a\) , \( 7 a - 18\) , \( -29 a + 66\bigr] \) ${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(7a-18\right){x}-29a+66$
729.1-g2 729.1-g \(\Q(\sqrt{13}) \) \( 3^{6} \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.682923374$ $6.472028930$ 2.451719308 \( 16755 a + 22482 \) \( \bigl[a\) , \( -a\) , \( 1\) , \( -11 a - 12\) , \( -20 a - 26\bigr] \) ${y}^2+a{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-11a-12\right){x}-20a-26$
729.1-h1 729.1-h \(\Q(\sqrt{13}) \) \( 3^{6} \) 0 $\Z/3\Z$ $-27$ $N(\mathrm{U}(1))$ $1$ $28.08911226$ 0.865613115 \( -12288000 \) \( \bigl[0\) , \( 0\) , \( 1\) , \( -30\) , \( 63\bigr] \) ${y}^2+{y}={x}^{3}-30{x}+63$
729.1-h2 729.1-h \(\Q(\sqrt{13}) \) \( 3^{6} \) 0 $\mathsf{trivial}$ $-27$ $N(\mathrm{U}(1))$ $1$ $0.346779163$ 0.865613115 \( -12288000 \) \( \bigl[0\) , \( 0\) , \( 1\) , \( -270\) , \( -1708\bigr] \) ${y}^2+{y}={x}^{3}-270{x}-1708$
729.1-h3 729.1-h \(\Q(\sqrt{13}) \) \( 3^{6} \) 0 $\Z/3\Z$ $-3$ $N(\mathrm{U}(1))$ $1$ $28.08911226$ 0.865613115 \( 0 \) \( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( 0\bigr] \) ${y}^2+{y}={x}^{3}$
729.1-h4 729.1-h \(\Q(\sqrt{13}) \) \( 3^{6} \) 0 $\Z/3\Z$ $-3$ $N(\mathrm{U}(1))$ $1$ $3.121012474$ 0.865613115 \( 0 \) \( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( -7\bigr] \) ${y}^2+{y}={x}^{3}-7$
729.1-i1 729.1-i \(\Q(\sqrt{13}) \) \( 3^{6} \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.354151165$ $14.41546786$ 2.831885807 \( -13515 a + 30222 \) \( \bigl[a\) , \( -a\) , \( a\) , \( -2 a - 3\) , \( -2 a - 3\bigr] \) ${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-2a-3\right){x}-2a-3$
729.1-i2 729.1-i \(\Q(\sqrt{13}) \) \( 3^{6} \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.118050388$ $14.41546786$ 2.831885807 \( -16755 a + 39237 \) \( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( 3 a + 4\) , \( 10 a + 13\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(3a+4\right){x}+10a+13$
729.1-j1 729.1-j \(\Q(\sqrt{13}) \) \( 3^{6} \) $1$ $\Z/3\Z$ $\mathrm{SU}(2)$ $0.227641124$ $19.41608679$ 2.451719308 \( -13515 a + 30222 \) \( \bigl[1\) , \( -1\) , \( a + 1\) , \( -8 a - 11\) , \( 28 a + 37\bigr] \) ${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-8a-11\right){x}+28a+37$
729.1-j2 729.1-j \(\Q(\sqrt{13}) \) \( 3^{6} \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.682923374$ $6.472028930$ 2.451719308 \( -16755 a + 39237 \) \( \bigl[a + 1\) , \( -1\) , \( 1\) , \( 10 a - 23\) , \( 20 a - 46\bigr] \) ${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-{x}^{2}+\left(10a-23\right){x}+20a-46$
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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.