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SageMath
E = EllipticCurve("cb1")
E.isogeny_class()
Elliptic curves in class 317400.cb
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
317400.cb1 | 317400cb4 | \([0, 1, 0, -13970008, -19782002512]\) | \(63649751618/1164375\) | \(5515817224140000000000\) | \([2]\) | \(19464192\) | \(2.9667\) | |
317400.cb2 | 317400cb2 | \([0, 1, 0, -1803008, 463885488]\) | \(273671716/119025\) | \(281919547011600000000\) | \([2, 2]\) | \(9732096\) | \(2.6201\) | |
317400.cb3 | 317400cb1 | \([0, 1, 0, -1538508, 733675488]\) | \(680136784/345\) | \(204289526820000000\) | \([2]\) | \(4866048\) | \(2.2735\) | \(\Gamma_0(N)\)-optimal |
317400.cb4 | 317400cb3 | \([0, 1, 0, 6131992, 3447445488]\) | \(5382838942/4197615\) | \(-19884725382551520000000\) | \([2]\) | \(19464192\) | \(2.9667\) |
Rank
sage: E.rank()
The elliptic curves in class 317400.cb have rank \(1\).
Complex multiplication
The elliptic curves in class 317400.cb do not have complex multiplication.Modular form 317400.2.a.cb
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.