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SageMath
E = EllipticCurve("pd1")
E.isogeny_class()
Elliptic curves in class 486720pd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
486720.pd1 | 486720pd1 | \([0, 0, 0, -132327, -16846596]\) | \(42144192/4225\) | \(25689644450308800\) | \([2]\) | \(3096576\) | \(1.8850\) | \(\Gamma_0(N)\)-optimal |
486720.pd2 | 486720pd2 | \([0, 0, 0, 164268, -81622944]\) | \(1259712/8125\) | \(-3161802393884160000\) | \([2]\) | \(6193152\) | \(2.2316\) |
Rank
sage: E.rank()
The elliptic curves in class 486720pd have rank \(0\).
Complex multiplication
The elliptic curves in class 486720pd do not have complex multiplication.Modular form 486720.2.a.pd
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.