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SageMath
E = EllipticCurve("co1")
E.isogeny_class()
Elliptic curves in class 75712.co
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
75712.co1 | 75712m1 | \([0, -1, 0, -7349697, 7672758209]\) | \(-1214950633/196\) | \(-7083200191503794176\) | \([]\) | \(3594240\) | \(2.6265\) | \(\Gamma_0(N)\)-optimal |
75712.co2 | 75712m2 | \([0, -1, 0, 1789823, 25079888001]\) | \(17546087/7529536\) | \(-272108218556809757065216\) | \([]\) | \(10782720\) | \(3.1758\) |
Rank
sage: E.rank()
The elliptic curves in class 75712.co have rank \(1\).
Complex multiplication
The elliptic curves in class 75712.co do not have complex multiplication.Modular form 75712.2.a.co
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.