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SageMath
E = EllipticCurve("eo1")
E.isogeny_class()
Elliptic curves in class 382200.eo
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
382200.eo1 | 382200eo4 | \([0, -1, 0, -10192408, -12521187188]\) | \(31103978031362/195\) | \(734129760000000\) | \([2]\) | \(10616832\) | \(2.4581\) | |
382200.eo2 | 382200eo3 | \([0, -1, 0, -882408, -31087188]\) | \(20183398562/11567205\) | \(43547843233440000000\) | \([2]\) | \(10616832\) | \(2.4581\) | |
382200.eo3 | 382200eo2 | \([0, -1, 0, -637408, -195237188]\) | \(15214885924/38025\) | \(71577651600000000\) | \([2, 2]\) | \(5308416\) | \(2.1115\) | |
382200.eo4 | 382200eo1 | \([0, -1, 0, -24908, -5362188]\) | \(-3631696/24375\) | \(-11470777500000000\) | \([2]\) | \(2654208\) | \(1.7649\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 382200.eo have rank \(1\).
Complex multiplication
The elliptic curves in class 382200.eo do not have complex multiplication.Modular form 382200.2.a.eo
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.