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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 304200e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
304200.e2 | 304200e1 | \([0, 0, 0, -63375, 13731250]\) | \(-432\) | \(-65161921500000000\) | \([2]\) | \(2764800\) | \(1.9147\) | \(\Gamma_0(N)\)-optimal |
304200.e1 | 304200e2 | \([0, 0, 0, -1330875, 590443750]\) | \(1000188\) | \(260647686000000000\) | \([2]\) | \(5529600\) | \(2.2613\) |
Rank
sage: E.rank()
The elliptic curves in class 304200e have rank \(1\).
Complex multiplication
The elliptic curves in class 304200e do not have complex multiplication.Modular form 304200.2.a.e
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.