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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 31850.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
31850.e1 | 31850be2 | \([1, 0, 1, -555196, 159180858]\) | \(-6434774386429585/140608\) | \(-413559764800\) | \([]\) | \(252720\) | \(1.7553\) | |
31850.e2 | 31850be1 | \([1, 0, 1, -6396, 248378]\) | \(-9836106385/3407872\) | \(-10023318323200\) | \([]\) | \(84240\) | \(1.2060\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 31850.e have rank \(1\).
Complex multiplication
The elliptic curves in class 31850.e do not have complex multiplication.Modular form 31850.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.