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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 3185.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3185.e1 | 3185i1 | \([1, 1, 1, -50, -50]\) | \(117649/65\) | \(7647185\) | \([2]\) | \(576\) | \(0.011340\) | \(\Gamma_0(N)\)-optimal |
3185.e2 | 3185i2 | \([1, 1, 1, 195, -148]\) | \(6967871/4225\) | \(-497067025\) | \([2]\) | \(1152\) | \(0.35791\) |
Rank
sage: E.rank()
The elliptic curves in class 3185.e have rank \(0\).
Complex multiplication
The elliptic curves in class 3185.e do not have complex multiplication.Modular form 3185.2.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.