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SageMath
E = EllipticCurve("ej1")
E.isogeny_class()
Elliptic curves in class 283920ej
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
283920.ej4 | 283920ej1 | \([0, 1, 0, -29631, -1973100]\) | \(37256083456/525\) | \(40545195600\) | \([2]\) | \(589824\) | \(1.1757\) | \(\Gamma_0(N)\)-optimal |
283920.ej3 | 283920ej2 | \([0, 1, 0, -30476, -1855476]\) | \(2533446736/275625\) | \(340579643040000\) | \([2, 2]\) | \(1179648\) | \(1.5222\) | |
283920.ej2 | 283920ej3 | \([0, 1, 0, -114976, 12982724]\) | \(34008619684/4862025\) | \(24031299612902400\) | \([2, 2]\) | \(2359296\) | \(1.8688\) | |
283920.ej5 | 283920ej4 | \([0, 1, 0, 40504, -9152220]\) | \(1486779836/8203125\) | \(-40545195600000000\) | \([2]\) | \(2359296\) | \(1.8688\) | |
283920.ej1 | 283920ej5 | \([0, 1, 0, -1771176, 906668244]\) | \(62161150998242/1607445\) | \(15890083825674240\) | \([2]\) | \(4718592\) | \(2.2154\) | |
283920.ej6 | 283920ej6 | \([0, 1, 0, 189224, 70294004]\) | \(75798394558/259416045\) | \(-2564406683136829440\) | \([2]\) | \(4718592\) | \(2.2154\) |
Rank
sage: E.rank()
The elliptic curves in class 283920ej have rank \(2\).
Complex multiplication
The elliptic curves in class 283920ej do not have complex multiplication.Modular form 283920.2.a.ej
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.