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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 7245.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
7245.e1 | 7245c1 | \([1, -1, 1, -37343, 2716606]\) | \(292583028222603/8456021875\) | \(166439878565625\) | \([2]\) | \(25920\) | \(1.5067\) | \(\Gamma_0(N)\)-optimal |
7245.e2 | 7245c2 | \([1, -1, 1, 8962, 8977042]\) | \(4044759171237/1771943359375\) | \(-34877161142578125\) | \([2]\) | \(51840\) | \(1.8533\) |
Rank
sage: E.rank()
The elliptic curves in class 7245.e have rank \(0\).
Complex multiplication
The elliptic curves in class 7245.e do not have complex multiplication.Modular form 7245.2.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.