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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 75712g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
75712.bb2 | 75712g1 | \([0, -1, 0, 191, -703]\) | \(17303/14\) | \(-620232704\) | \([]\) | \(27648\) | \(0.37498\) | \(\Gamma_0(N)\)-optimal |
75712.bb1 | 75712g2 | \([0, -1, 0, -3969, -96383]\) | \(-156116857/2744\) | \(-121565609984\) | \([]\) | \(82944\) | \(0.92428\) |
Rank
sage: E.rank()
The elliptic curves in class 75712g have rank \(1\).
Complex multiplication
The elliptic curves in class 75712g do not have complex multiplication.Modular form 75712.2.a.g
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 Cremona 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 Cremona labels.