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SageMath
E = EllipticCurve("di1")
E.isogeny_class()
Elliptic curves in class 96330di
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
96330.de4 | 96330di1 | \([1, 0, 0, 621325, -1131315615]\) | \(5495662324535111/117739817533440\) | \(-568307610928765992960\) | \([2]\) | \(5160960\) | \(2.6627\) | \(\Gamma_0(N)\)-optimal |
96330.de3 | 96330di2 | \([1, 0, 0, -13223155, -17531486623]\) | \(52974743974734147769/3152005008998400\) | \(15214126145478558105600\) | \([2, 2]\) | \(10321920\) | \(3.0093\) | |
96330.de2 | 96330di3 | \([1, 0, 0, -39506035, 73791008225]\) | \(1412712966892699019449/330160465517040000\) | \(1593621506401838325360000\) | \([2]\) | \(20643840\) | \(3.3559\) | |
96330.de1 | 96330di4 | \([1, 0, 0, -208451955, -1158409548063]\) | \(207530301091125281552569/805586668007040\) | \(3888412979416392735360\) | \([2]\) | \(20643840\) | \(3.3559\) |
Rank
sage: E.rank()
The elliptic curves in class 96330di have rank \(1\).
Complex multiplication
The elliptic curves in class 96330di do not have complex multiplication.Modular form 96330.2.a.di
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.