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## Results (36 matches)

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Label Class Base field Conductor norm Rank Torsion CM Weierstrass equation
17500.2-a1 17500.2-a $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $1$ $\mathsf{trivial}$ ${y}^2+{x}{y}={x}^{3}+{x}^{2}-45{x}-185$
17500.2-a2 17500.2-a $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $1$ $\mathsf{trivial}$ ${y}^2+{x}{y}={x}^{3}+{x}^{2}+5{x}+5$
17500.2-b1 17500.2-b $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $0$ $\Z/2\Z$ ${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(-150a-17\right){x}-900a+641$
17500.2-b2 17500.2-b $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $0$ $\Z/2\Z$ ${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(150a-167\right){x}+900a-259$
17500.2-b3 17500.2-b $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $0$ $\Z/2\Z\oplus\Z/2\Z$ ${y}^2+{x}{y}={x}^{3}-{x}^{2}+58{x}-284$
17500.2-b4 17500.2-b $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $0$ $\Z/2\Z\oplus\Z/2\Z$ ${y}^2+{x}{y}={x}^{3}-{x}^{2}-442{x}-2784$
17500.2-b5 17500.2-b $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $0$ $\Z/2\Z$ ${y}^2+{x}{y}={x}^{3}-{x}^{2}-2192{x}+37466$
17500.2-b6 17500.2-b $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $0$ $\Z/2\Z$ ${y}^2+{x}{y}={x}^{3}-{x}^{2}-6692{x}-209034$
17500.2-c1 17500.2-c $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $0$ $\mathsf{trivial}$ ${y}^2+{x}{y}={x}^{3}-{x}^{2}-4492{x}+126416$
17500.2-d1 17500.2-d $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $0$ $\Z/2\Z$ ${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(87a+2991\right){x}-55657a+43491$
17500.2-d2 17500.2-d $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $0$ $\Z/2\Z$ ${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(837a+1749\right){x}-41832a+53336$
17500.2-d3 17500.2-d $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $0$ $\Z/2\Z$ ${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-38a-259\right){x}+468a-1259$
17500.2-d4 17500.2-d $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $0$ $\Z/2\Z$ ${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-47288a+36749\right){x}-2146207a+6589586$
17500.2-d5 17500.2-d $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $0$ $\Z/2\Z$ ${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(712a-501\right){x}-7207a-2414$
17500.2-d6 17500.2-d $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $0$ $\Z/2\Z$ ${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(15712a+246741\right){x}-37365032a+24812241$
17500.2-e1 17500.2-e $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $0$ $\Z/2\Z$ ${y}^2+{x}{y}+a{y}={x}^{3}+\left(-713a+212\right){x}+7206a-9620$
17500.2-e2 17500.2-e $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $0$ $\Z/2\Z$ ${y}^2+{x}{y}+a{y}={x}^{3}+\left(-838a+2587\right){x}+41831a+11505$
17500.2-e3 17500.2-e $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $0$ $\Z/2\Z$ ${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-87a+3078\right){x}+55657a-12166$
17500.2-e4 17500.2-e $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $0$ $\Z/2\Z$ ${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(38a-297\right){x}-468a-791$
17500.2-e5 17500.2-e $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $0$ $\Z/2\Z$ ${y}^2+{x}{y}+a{y}={x}^{3}+\left(47287a-10538\right){x}+2146206a+4443380$
17500.2-e6 17500.2-e $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $0$ $\Z/2\Z$ ${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-15712a+262453\right){x}+37365032a-12552791$
17500.2-f1 17500.2-f $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $1$ $\Z/2\Z\oplus\Z/2\Z$ ${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-4263{x}-109219$
17500.2-f2 17500.2-f $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $1$ $\Z/2\Z$ ${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(250a+153\right){x}+838a-3853$
17500.2-f3 17500.2-f $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $1$ $\Z/2\Z$ ${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-251a+403\right){x}-839a-3015$
17500.2-f4 17500.2-f $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $1$ $\Z/2\Z$ ${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-a+28\right){x}+36a-15$
17500.2-f5 17500.2-f $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $1$ $\Z/2\Z$ ${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+28{x}-37a+22$
17500.2-f6 17500.2-f $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $1$ $\Z/2\Z\oplus\Z/2\Z$ ${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-13{x}+31$
17500.2-f7 17500.2-f $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $1$ $\Z/2\Z\oplus\Z/2\Z$ ${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+112{x}-719$
17500.2-f8 17500.2-f $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $1$ $\Z/2\Z$ ${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-750a-222\right){x}+3213a-19478$
17500.2-f9 17500.2-f $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $1$ $\Z/2\Z$ ${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(749a-972\right){x}-3214a-16265$
17500.2-f10 17500.2-f $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $1$ $\Z/2\Z$ ${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-888{x}-8719$
17500.2-f11 17500.2-f $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $1$ $\Z/2\Z$ ${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-263{x}+1531$
17500.2-f12 17500.2-f $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $1$ $\Z/2\Z$ ${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-68263{x}-6893219$
17500.2-g1 17500.2-g $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $1$ $\mathsf{trivial}$ ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-180{x}+1047$
17500.2-h1 17500.2-h $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $0$ $\mathsf{trivial}$ ${y}^2+{x}{y}={x}^{3}-1138{x}-20858$
17500.2-h2 17500.2-h $$\Q(\sqrt{-7})$$ $$2^{2} \cdot 5^{4} \cdot 7$$ $0$ $\Z/3\Z$ ${y}^2+{x}{y}={x}^{3}+112{x}+392$
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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.