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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 458640.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
458640.e1 | 458640e1 | \([0, 0, 0, -35868, -250733]\) | \(3718856704/2132325\) | \(2926099903381200\) | \([2]\) | \(2211840\) | \(1.6574\) | \(\Gamma_0(N)\)-optimal |
458640.e2 | 458640e2 | \([0, 0, 0, 142737, -2001062]\) | \(14647977776/8555625\) | \(-187848388859040000\) | \([2]\) | \(4423680\) | \(2.0040\) |
Rank
sage: E.rank()
The elliptic curves in class 458640.e have rank \(1\).
Complex multiplication
The elliptic curves in class 458640.e do not have complex multiplication.Modular form 458640.2.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.