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SageMath
E = EllipticCurve("db1")
E.isogeny_class()
Elliptic curves in class 463680.db
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
463680.db1 | 463680db3 | \([0, 0, 0, -2473068, 1496932432]\) | \(8753151307882969/65205\) | \(12460869550080\) | \([2]\) | \(5767168\) | \(2.1074\) | \(\Gamma_0(N)\)-optimal* |
463680.db2 | 463680db2 | \([0, 0, 0, -154668, 23357392]\) | \(2141202151369/5832225\) | \(1114555554201600\) | \([2, 2]\) | \(2883584\) | \(1.7608\) | \(\Gamma_0(N)\)-optimal* |
463680.db3 | 463680db4 | \([0, 0, 0, -94188, 41815888]\) | \(-483551781049/3672913125\) | \(-701904628776960000\) | \([2]\) | \(5767168\) | \(2.1074\) | |
463680.db4 | 463680db1 | \([0, 0, 0, -13548, 44368]\) | \(1439069689/828345\) | \(158299194654720\) | \([2]\) | \(1441792\) | \(1.4142\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 463680.db have rank \(2\).
Complex multiplication
The elliptic curves in class 463680.db do not have complex multiplication.Modular form 463680.2.a.db
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.