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SageMath
E = EllipticCurve("fr1")
E.isogeny_class()
Elliptic curves in class 136710.fr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
136710.fr1 | 136710g2 | \([1, -1, 1, -96335357, 363961474629]\) | \(1152829477932246539641/3188367360\) | \(273453900790210560\) | \([2]\) | \(9584640\) | \(3.0040\) | |
136710.fr2 | 136710g1 | \([1, -1, 1, -6018557, 5692792389]\) | \(-281115640967896441/468084326400\) | \(-40145776976225894400\) | \([2]\) | \(4792320\) | \(2.6574\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 136710.fr have rank \(1\).
Complex multiplication
The elliptic curves in class 136710.fr do not have complex multiplication.Modular form 136710.2.a.fr
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.