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## Results (24 matches)

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Label Class Base field Conductor norm Rank Torsion CM Weierstrass equation
686.1-a1 686.1-a $$\Q(\sqrt{2})$$ $$2 \cdot 7^{3}$$ $0$ $\Z/2\Z$ ${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(43a-6\right){x}+65a-135$
686.1-a2 686.1-a $$\Q(\sqrt{2})$$ $$2 \cdot 7^{3}$$ $0$ $\Z/2\Z$ ${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(303a-646\right){x}+4929a-6831$
686.1-b1 686.1-b $$\Q(\sqrt{2})$$ $$2 \cdot 7^{3}$$ $1$ $\Z/3\Z$ ${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(5a-9\right){x}-15a+21$
686.1-b2 686.1-b $$\Q(\sqrt{2})$$ $$2 \cdot 7^{3}$$ $1$ $\mathsf{trivial}$ ${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-25a+46\right){x}-20a+22$
686.1-c1 686.1-c $$\Q(\sqrt{2})$$ $$2 \cdot 7^{3}$$ $1$ $\mathsf{trivial}$ ${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-31a-50\right){x}+121a+124$
686.1-c2 686.1-c $$\Q(\sqrt{2})$$ $$2 \cdot 7^{3}$$ $1$ $\mathsf{trivial}$ ${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}-a{x}-a$
686.1-d1 686.1-d $$\Q(\sqrt{2})$$ $$2 \cdot 7^{3}$$ $0$ $\Z/4\Z$ ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-683a-1536\right){x}+19222a+21843$
686.1-d2 686.1-d $$\Q(\sqrt{2})$$ $$2 \cdot 7^{3}$$ $0$ $\Z/4\Z$ ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-3a-6\right){x}-6a-7$
686.1-d3 686.1-d $$\Q(\sqrt{2})$$ $$2 \cdot 7^{3}$$ $0$ $\Z/4\Z$ ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(17a+39\right){x}+126a+143$
686.1-d4 686.1-d $$\Q(\sqrt{2})$$ $$2 \cdot 7^{3}$$ $0$ $\Z/2\Z$ ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(132a-376\right){x}-1908a+2353$
686.1-d5 686.1-d $$\Q(\sqrt{2})$$ $$2 \cdot 7^{3}$$ $0$ $\Z/2\Z\oplus\Z/2\Z$ ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-143a-321\right){x}+1534a+1743$
686.1-d6 686.1-d $$\Q(\sqrt{2})$$ $$2 \cdot 7^{3}$$ $0$ $\Z/2\Z$ ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-253a-2161\right){x}-5170a-37057$
686.1-d7 686.1-d $$\Q(\sqrt{2})$$ $$2 \cdot 7^{3}$$ $0$ $\Z/2\Z$ ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(37477a-93056\right){x}+8696054a-9208877$
686.1-d8 686.1-d $$\Q(\sqrt{2})$$ $$2 \cdot 7^{3}$$ $0$ $\Z/2\Z\oplus\Z/2\Z$ ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-43a-96\right){x}-270a-307$
686.1-d9 686.1-d $$\Q(\sqrt{2})$$ $$2 \cdot 7^{3}$$ $0$ $\Z/2\Z$ ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-2593a-4241\right){x}+94830a+138943$
686.1-d10 686.1-d $$\Q(\sqrt{2})$$ $$2 \cdot 7^{3}$$ $0$ $\Z/2\Z\oplus\Z/2\Z$ ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-10923a-24576\right){x}+1213206a+1378643$
686.1-d11 686.1-d $$\Q(\sqrt{2})$$ $$2 \cdot 7^{3}$$ $0$ $\Z/2\Z$ ${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(485a-759\right){x}-2705a+3591$
686.1-d12 686.1-d $$\Q(\sqrt{2})$$ $$2 \cdot 7^{3}$$ $0$ $\Z/2\Z$ ${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(103060a-164599\right){x}+23909933a-32816619$
686.1-e1 686.1-e $$\Q(\sqrt{2})$$ $$2 \cdot 7^{3}$$ $0$ $\mathsf{trivial}$ ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-476a-696\right){x}-7429a-10807$
686.1-e2 686.1-e $$\Q(\sqrt{2})$$ $$2 \cdot 7^{3}$$ $0$ $\Z/3\Z$ ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-6a-6\right){x}-21a-19$
686.1-f1 686.1-f $$\Q(\sqrt{2})$$ $$2 \cdot 7^{3}$$ $0$ $\Z/2\Z$ ${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(9a-7\right){x}+16a-19$
686.1-f2 686.1-f $$\Q(\sqrt{2})$$ $$2 \cdot 7^{3}$$ $0$ $\Z/2\Z$ ${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-176a-256\right){x}+36a+32$
686.1-g1 686.1-g $$\Q(\sqrt{2})$$ $$2 \cdot 7^{3}$$ $0$ $\mathsf{trivial}$ ${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(13a-39\right){x}-67a+51$
686.1-g2 686.1-g $$\Q(\sqrt{2})$$ $$2 \cdot 7^{3}$$ $0$ $\mathsf{trivial}$ ${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-37a+216\right){x}-64a+756$
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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.