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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 121968.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
121968.e1 | 121968fi2 | \([0, 0, 0, -21747, 1061170]\) | \(286191179/43218\) | \(171763229417472\) | \([2]\) | \(589824\) | \(1.4555\) | |
121968.e2 | 121968fi1 | \([0, 0, 0, -5907, -158510]\) | \(5735339/588\) | \(2336914685952\) | \([2]\) | \(294912\) | \(1.1089\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 121968.e have rank \(2\).
Complex multiplication
The elliptic curves in class 121968.e do not have complex multiplication.Modular form 121968.2.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 LMFDB 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 LMFDB labels.