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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 4560q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4560.a4 | 4560q1 | \([0, -1, 0, 58824, 32937840]\) | \(5495662324535111/117739817533440\) | \(-482262292616970240\) | \([2]\) | \(53760\) | \(2.0734\) | \(\Gamma_0(N)\)-optimal |
4560.a3 | 4560q2 | \([0, -1, 0, -1251896, 511088496]\) | \(52974743974734147769/3152005008998400\) | \(12910612516857446400\) | \([2, 2]\) | \(107520\) | \(2.4200\) | |
4560.a2 | 4560q3 | \([0, -1, 0, -3740216, -2148427920]\) | \(1412712966892699019449/330160465517040000\) | \(1352337266757795840000\) | \([2]\) | \(215040\) | \(2.7666\) | |
4560.a1 | 4560q4 | \([0, -1, 0, -19735096, 33751275376]\) | \(207530301091125281552569/805586668007040\) | \(3299682992156835840\) | \([2]\) | \(215040\) | \(2.7666\) |
Rank
sage: E.rank()
The elliptic curves in class 4560q have rank \(1\).
Complex multiplication
The elliptic curves in class 4560q do not have complex multiplication.Modular form 4560.2.a.q
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.