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SageMath
E = EllipticCurve("ff1")
E.isogeny_class()
Elliptic curves in class 463680ff
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
463680.ff2 | 463680ff1 | \([0, 0, 0, -512748, -191690928]\) | \(-78013216986489/37918720000\) | \(-7246380238110720000\) | \([2]\) | \(8257536\) | \(2.3244\) | \(\Gamma_0(N)\)-optimal* |
463680.ff1 | 463680ff2 | \([0, 0, 0, -8991468, -10376329392]\) | \(420676324562824569/56350000000\) | \(10768652697600000000\) | \([2]\) | \(16515072\) | \(2.6710\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 463680ff have rank \(0\).
Complex multiplication
The elliptic curves in class 463680ff do not have complex multiplication.Modular form 463680.2.a.ff
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.