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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 5525e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5525.c1 | 5525e1 | \([1, 0, 0, -1488, 21967]\) | \(23320116793/2873\) | \(44890625\) | \([2]\) | \(3072\) | \(0.49170\) | \(\Gamma_0(N)\)-optimal |
5525.c2 | 5525e2 | \([1, 0, 0, -1363, 25842]\) | \(-17923019113/8254129\) | \(-128970765625\) | \([2]\) | \(6144\) | \(0.83827\) |
Rank
sage: E.rank()
The elliptic curves in class 5525e have rank \(2\).
Complex multiplication
The elliptic curves in class 5525e do not have complex multiplication.Modular form 5525.2.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.