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SageMath
E = EllipticCurve("bs1")
E.isogeny_class()
Elliptic curves in class 463680bs
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
463680.bs4 | 463680bs1 | \([0, 0, 0, 792, -20088]\) | \(73598976/276115\) | \(-206118743040\) | \([2]\) | \(327680\) | \(0.85380\) | \(\Gamma_0(N)\)-optimal* |
463680.bs3 | 463680bs2 | \([0, 0, 0, -8028, -242352]\) | \(4790692944/648025\) | \(7739969126400\) | \([2, 2]\) | \(655360\) | \(1.2004\) | \(\Gamma_0(N)\)-optimal* |
463680.bs2 | 463680bs3 | \([0, 0, 0, -33228, 2086128]\) | \(84923690436/9794435\) | \(467936419184640\) | \([2]\) | \(1310720\) | \(1.5469\) | \(\Gamma_0(N)\)-optimal* |
463680.bs1 | 463680bs4 | \([0, 0, 0, -123948, -16795728]\) | \(4407931365156/100625\) | \(4807434240000\) | \([2]\) | \(1310720\) | \(1.5469\) |
Rank
sage: E.rank()
The elliptic curves in class 463680bs have rank \(2\).
Complex multiplication
The elliptic curves in class 463680bs do not have complex multiplication.Modular form 463680.2.a.bs
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.