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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 3450.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3450.c1 | 3450a2 | \([1, 1, 0, -19650, -742500]\) | \(53706380371489/16171875000\) | \(252685546875000\) | \([2]\) | \(11520\) | \(1.4688\) | |
3450.c2 | 3450a1 | \([1, 1, 0, 3350, -75500]\) | \(265971760991/317400000\) | \(-4959375000000\) | \([2]\) | \(5760\) | \(1.1222\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3450.c have rank \(1\).
Complex multiplication
The elliptic curves in class 3450.c do not have complex multiplication.Modular form 3450.2.a.c
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.