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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 32912o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
32912.c2 | 32912o1 | \([0, 1, 0, -1008, -9308]\) | \(62500/17\) | \(30839333888\) | \([2]\) | \(22400\) | \(0.72053\) | \(\Gamma_0(N)\)-optimal |
32912.c1 | 32912o2 | \([0, 1, 0, -5848, 162996]\) | \(6097250/289\) | \(1048537352192\) | \([2]\) | \(44800\) | \(1.0671\) |
Rank
sage: E.rank()
The elliptic curves in class 32912o have rank \(1\).
Complex multiplication
The elliptic curves in class 32912o do not have complex multiplication.Modular form 32912.2.a.o
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.