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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 8712g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8712.z2 | 8712g1 | \([0, 0, 0, -8283, 289190]\) | \(63253004/243\) | \(241441440768\) | \([2]\) | \(23040\) | \(1.0426\) | \(\Gamma_0(N)\)-optimal |
8712.z1 | 8712g2 | \([0, 0, 0, -12243, -15730]\) | \(102129622/59049\) | \(117340540213248\) | \([2]\) | \(46080\) | \(1.3892\) |
Rank
sage: E.rank()
The elliptic curves in class 8712g have rank \(0\).
Complex multiplication
The elliptic curves in class 8712g do not have complex multiplication.Modular form 8712.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.