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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 4275.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4275.e1 | 4275q2 | \([1, -1, 1, -5555, -154178]\) | \(13312053/361\) | \(514001953125\) | \([2]\) | \(5120\) | \(1.0274\) | |
4275.e2 | 4275q1 | \([1, -1, 1, 70, -7928]\) | \(27/19\) | \(-27052734375\) | \([2]\) | \(2560\) | \(0.68079\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4275.e have rank \(0\).
Complex multiplication
The elliptic curves in class 4275.e do not have complex multiplication.Modular form 4275.2.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.