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## Results (1-50 of 10611 matches)

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Label Class Base field Conductor norm Rank Torsion CM Weierstrass equation
10.1-a1 10.1-a $$\Q(\sqrt{-31})$$ $$2 \cdot 5$$ $0$ $\Z/2\Z$ ${y}^2+a{x}{y}+a{y}={x}^3+\left(a-1\right){x}^2+\left(-67a+360\right){x}+905a+1396$
10.1-a2 10.1-a $$\Q(\sqrt{-31})$$ $$2 \cdot 5$$ $0$ $\Z/2\Z$ ${y}^2+a{x}{y}+a{y}={x}^3+\left(a-1\right){x}^2-2a{x}-a+4$
10.1-a3 10.1-a $$\Q(\sqrt{-31})$$ $$2 \cdot 5$$ $0$ $\Z/2\Z$ ${y}^2+{x}{y}+{y}={x}^3+\left(-a-1\right){x}^2+\left(a-3\right){x}+1$
10.1-a4 10.1-a $$\Q(\sqrt{-31})$$ $$2 \cdot 5$$ $0$ $\Z/2\Z$ ${y}^2+{x}{y}+{y}={x}^3+\left(-a-1\right){x}^2+\left(-4a+17\right){x}+12a-1$
10.4-a1 10.4-a $$\Q(\sqrt{-31})$$ $$2 \cdot 5$$ $0$ $\Z/2\Z$ ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+a{x}^2+\left(62a+289\right){x}-550a+1785$
10.4-a2 10.4-a $$\Q(\sqrt{-31})$$ $$2 \cdot 5$$ $0$ $\Z/2\Z$ ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+a{x}^2+\left(-3a-6\right){x}-4a+7$
10.4-a3 10.4-a $$\Q(\sqrt{-31})$$ $$2 \cdot 5$$ $0$ $\Z/2\Z$ ${y}^2+{x}{y}={x}^3+\left(a+1\right){x}^2+\left(a-2\right){x}-1$
10.4-a4 10.4-a $$\Q(\sqrt{-31})$$ $$2 \cdot 5$$ $0$ $\Z/2\Z$ ${y}^2+{x}{y}={x}^3+\left(a+1\right){x}^2+\left(6a+13\right){x}-7a+24$
14.2-a1 14.2-a $$\Q(\sqrt{-31})$$ $$2 \cdot 7$$ $0$ $\Z/3\Z$ ${y}^2+\left(a+1\right){x}{y}+{y}={x}^3+\left(-a+29\right){x}-24a-42$
14.2-a2 14.2-a $$\Q(\sqrt{-31})$$ $$2 \cdot 7$$ $0$ $\mathsf{trivial}$ ${y}^2+\left(a+1\right){x}{y}+{y}={x}^3+\left(4a-266\right){x}+467a+1516$
14.2-a3 14.2-a $$\Q(\sqrt{-31})$$ $$2 \cdot 7$$ $0$ $\Z/3\Z$ ${y}^2+\left(a+1\right){x}{y}+{y}={x}^3+\left(-a-1\right){x}$
14.2-a4 14.2-a $$\Q(\sqrt{-31})$$ $$2 \cdot 7$$ $0$ $\Z/3\Z$ ${y}^2+\left(a+1\right){x}{y}+{y}={x}^3+\left(4a+4\right){x}-a-38$
14.3-a1 14.3-a $$\Q(\sqrt{-31})$$ $$2 \cdot 7$$ $0$ $\Z/3\Z$ ${y}^2+a{x}{y}+{y}={x}^3+\left(-a+1\right){x}^2+\left(-5a+9\right){x}+a-39$
14.3-a2 14.3-a $$\Q(\sqrt{-31})$$ $$2 \cdot 7$$ $0$ $\Z/3\Z$ ${y}^2+a{x}{y}+{y}={x}^3+\left(-a+1\right){x}^2+29{x}+24a-66$
14.3-a3 14.3-a $$\Q(\sqrt{-31})$$ $$2 \cdot 7$$ $0$ $\mathsf{trivial}$ ${y}^2+a{x}{y}+{y}={x}^3+\left(-a+1\right){x}^2+\left(-5a-261\right){x}-467a+1983$
14.3-a4 14.3-a $$\Q(\sqrt{-31})$$ $$2 \cdot 7$$ $0$ $\Z/3\Z$ ${y}^2+a{x}{y}+{y}={x}^3+\left(-a+1\right){x}^2-{x}$
20.3-a1 20.3-a $$\Q(\sqrt{-31})$$ $$2^{2} \cdot 5$$ $1$ $\Z/2\Z$ ${y}^2+a{x}{y}+a{y}={x}^3+\left(-a+1\right){x}^2+\left(4a+40\right){x}-38a+48$
20.3-a2 20.3-a $$\Q(\sqrt{-31})$$ $$2^{2} \cdot 5$$ $1$ $\Z/6\Z$ ${y}^2+a{x}{y}+a{y}={x}^3+\left(-a+1\right){x}^2-a{x}$
20.3-a3 20.3-a $$\Q(\sqrt{-31})$$ $$2^{2} \cdot 5$$ $1$ $\Z/6\Z$ ${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(-4a-4\right){x}+16$
20.3-a4 20.3-a $$\Q(\sqrt{-31})$$ $$2^{2} \cdot 5$$ $1$ $\Z/2\Z$ ${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(11a+21\right){x}-5a+45$
20.3-b1 20.3-b $$\Q(\sqrt{-31})$$ $$2^{2} \cdot 5$$ $0$ $\Z/2\Z$ ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-{x}^2+\left(78a-963\right){x}-1718a+12067$
20.3-b2 20.3-b $$\Q(\sqrt{-31})$$ $$2^{2} \cdot 5$$ $0$ $\Z/6\Z$ ${y}^2+a{x}{y}+a{y}={x}^3+\left(3a+29\right){x}+16a-58$
20.3-b3 20.3-b $$\Q(\sqrt{-31})$$ $$2^{2} \cdot 5$$ $0$ $\Z/6\Z$ ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-{x}^2+\left(-2a-3\right){x}-6a+35$
20.3-b4 20.3-b $$\Q(\sqrt{-31})$$ $$2^{2} \cdot 5$$ $0$ $\Z/2\Z$ ${y}^2+a{x}{y}+a{y}={x}^3+\left(68a-171\right){x}+452a-346$
20.4-a1 20.4-a $$\Q(\sqrt{-31})$$ $$2^{2} \cdot 5$$ $1$ $\Z/6\Z$ ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-a-1\right){x}-a$
20.4-a2 20.4-a $$\Q(\sqrt{-31})$$ $$2^{2} \cdot 5$$ $1$ $\Z/2\Z$ ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-6a+44\right){x}+37a+10$
20.4-a3 20.4-a $$\Q(\sqrt{-31})$$ $$2^{2} \cdot 5$$ $1$ $\Z/6\Z$ ${y}^2+a{x}{y}+a{y}={x}^3+\left(a+1\right){x}^2+a{x}$
20.4-a4 20.4-a $$\Q(\sqrt{-31})$$ $$2^{2} \cdot 5$$ $1$ $\Z/2\Z$ ${y}^2+a{x}{y}+a{y}={x}^3+\left(a+1\right){x}^2+\left(-14a+40\right){x}+30a+144$
20.4-b1 20.4-b $$\Q(\sqrt{-31})$$ $$2^{2} \cdot 5$$ $0$ $\Z/2\Z$ ${y}^2+a{x}{y}+a{y}={x}^3-a{x}^2+\left(-80a-883\right){x}+1717a+10350$
20.4-b2 20.4-b $$\Q(\sqrt{-31})$$ $$2^{2} \cdot 5$$ $0$ $\Z/2\Z$ ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(-70a-103\right){x}-453a+106$
20.4-b3 20.4-b $$\Q(\sqrt{-31})$$ $$2^{2} \cdot 5$$ $0$ $\Z/6\Z$ ${y}^2+a{x}{y}+a{y}={x}^3-a{x}^2-3{x}+5a+30$
20.4-b4 20.4-b $$\Q(\sqrt{-31})$$ $$2^{2} \cdot 5$$ $0$ $\Z/6\Z$ ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(-5a+32\right){x}-17a-42$
32.2-a1 32.2-a $$\Q(\sqrt{-31})$$ $$2^{5}$$ $0$ $\Z/2\Z$ ${y}^2+a{x}{y}+a{y}={x}^3+\left(-a+1\right){x}^2+\left(-2a+8\right){x}+4$
32.2-a2 32.2-a $$\Q(\sqrt{-31})$$ $$2^{5}$$ $0$ $\Z/2\Z$ ${y}^2+a{x}{y}+a{y}={x}^3+\left(2a+32\right){x}-20a+16$
32.2-a3 32.2-a $$\Q(\sqrt{-31})$$ $$2^{5}$$ $0$ $\Z/2\Z$ ${y}^2={x}^3-{x}^2+8{x}-4a+16$
32.2-a4 32.2-a $$\Q(\sqrt{-31})$$ $$2^{5}$$ $0$ $\Z/2\Z$ ${y}^2={x}^3+{x}^2+8{x}-4a-12$
32.3-a1 32.3-a $$\Q(\sqrt{-31})$$ $$2^{5}$$ $1$ $\mathsf{trivial}$ ${y}^2={x}^3-{x}^2+\left(-a+3\right){x}-3$
32.4-a1 32.4-a $$\Q(\sqrt{-31})$$ $$2^{5}$$ $1$ $\mathsf{trivial}$ ${y}^2={x}^3-{x}^2+\left(a+2\right){x}-3$
32.5-a1 32.5-a $$\Q(\sqrt{-31})$$ $$2^{5}$$ $0$ $\Z/2\Z$ ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(-4a+34\right){x}+19a-4$
32.5-a2 32.5-a $$\Q(\sqrt{-31})$$ $$2^{5}$$ $0$ $\Z/2\Z$ ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+6{x}-a+4$
32.5-a3 32.5-a $$\Q(\sqrt{-31})$$ $$2^{5}$$ $0$ $\Z/2\Z$ ${y}^2={x}^3+{x}^2+8{x}+4a-16$
32.5-a4 32.5-a $$\Q(\sqrt{-31})$$ $$2^{5}$$ $0$ $\Z/2\Z$ ${y}^2={x}^3-{x}^2+8{x}+4a+12$
49.1-a1 49.1-a $$\Q(\sqrt{-31})$$ $$7^{2}$$ $0$ $\mathsf{trivial}$ ${y}^2+a{x}{y}+\left(a+1\right){y}={x}^3+\left(-a-1\right){x}^2+\left(2a+2\right){x}-2a+7$
49.3-a1 49.3-a $$\Q(\sqrt{-31})$$ $$7^{2}$$ $0$ $\mathsf{trivial}$ ${y}^2+\left(a+1\right){x}{y}+{y}={x}^3+{x}^2+\left(-2a-4\right){x}-a+2$
50.1-a1 50.1-a $$\Q(\sqrt{-31})$$ $$2 \cdot 5^{2}$$ $0$ $\Z/2\Z$ ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a-1\right){x}^2+\left(-609a+1079\right){x}-674a+33097$
50.1-a2 50.1-a $$\Q(\sqrt{-31})$$ $$2 \cdot 5^{2}$$ $0$ $\Z/2\Z$ ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a-1\right){x}^2+\left(a-11\right){x}-9$
50.1-a3 50.1-a $$\Q(\sqrt{-31})$$ $$2 \cdot 5^{2}$$ $0$ $\Z/2\Z$ ${y}^2+a{x}{y}={x}^3+\left(-a+1\right){x}^2+\left(a+8\right){x}-8a$
50.1-a4 50.1-a $$\Q(\sqrt{-31})$$ $$2 \cdot 5^{2}$$ $0$ $\Z/2\Z$ ${y}^2+a{x}{y}={x}^3+\left(-a+1\right){x}^2+\left(-144a-112\right){x}+1415a-3512$
50.6-a1 50.6-a $$\Q(\sqrt{-31})$$ $$2 \cdot 5^{2}$$ $0$ $\Z/2\Z$ ${y}^2+a{x}{y}={x}^3+\left(a+1\right){x}^2+\left(605a+472\right){x}+1753a+27576$
50.6-a2 50.6-a $$\Q(\sqrt{-31})$$ $$2 \cdot 5^{2}$$ $0$ $\Z/2\Z$ ${y}^2+a{x}{y}={x}^3+\left(a+1\right){x}^2+\left(-5a-8\right){x}-11a+24$
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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.