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SageMath
E = EllipticCurve("cj1")
E.isogeny_class()
Elliptic curves in class 54720.cj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
54720.cj1 | 54720bl4 | \([0, 0, 0, -710463468, 7288854554288]\) | \(207530301091125281552569/805586668007040\) | \(153950009682069332951040\) | \([2]\) | \(13762560\) | \(3.6624\) | |
54720.cj2 | 54720bl3 | \([0, 0, 0, -134647788, -464329726288]\) | \(1412712966892699019449/330160465517040000\) | \(63094647517851722711040000\) | \([2]\) | \(13762560\) | \(3.6624\) | |
54720.cj3 | 54720bl2 | \([0, 0, 0, -45068268, 110304978608]\) | \(52974743974734147769/3152005008998400\) | \(602357537586501019238400\) | \([2, 2]\) | \(6881280\) | \(3.3159\) | |
54720.cj4 | 54720bl1 | \([0, 0, 0, 2117652, 7118808752]\) | \(5495662324535111/117739817533440\) | \(-22500429524337363517440\) | \([2]\) | \(3440640\) | \(2.9693\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 54720.cj have rank \(1\).
Complex multiplication
The elliptic curves in class 54720.cj do not have complex multiplication.Modular form 54720.2.a.cj
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.