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SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 494190.cm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
494190.cm1 | 494190cm2 | \([1, -1, 0, -78084, 8285490]\) | \(2992209121/54150\) | \(952838984424150\) | \([2]\) | \(3194880\) | \(1.6695\) | \(\Gamma_0(N)\)-optimal* |
494190.cm2 | 494190cm1 | \([1, -1, 0, -54, 373248]\) | \(-1/3420\) | \(-60179304279420\) | \([2]\) | \(1597440\) | \(1.3229\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 494190.cm have rank \(0\).
Complex multiplication
The elliptic curves in class 494190.cm do not have complex multiplication.Modular form 494190.2.a.cm
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.