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SageMath
E = EllipticCurve("cv1")
E.isogeny_class()
Elliptic curves in class 137904cv
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
137904.f3 | 137904cv1 | \([0, -1, 0, -8844, -316512]\) | \(61918288/153\) | \(189056454912\) | \([2]\) | \(245760\) | \(1.0416\) | \(\Gamma_0(N)\)-optimal |
137904.f2 | 137904cv2 | \([0, -1, 0, -12224, -48816]\) | \(40873252/23409\) | \(115702550406144\) | \([2, 2]\) | \(491520\) | \(1.3882\) | |
137904.f1 | 137904cv3 | \([0, -1, 0, -127144, 17419024]\) | \(22994537186/111537\) | \(1102577245046784\) | \([2]\) | \(983040\) | \(1.7348\) | |
137904.f4 | 137904cv4 | \([0, -1, 0, 48616, -438192]\) | \(1285471294/751689\) | \(-7430674903861248\) | \([2]\) | \(983040\) | \(1.7348\) |
Rank
sage: E.rank()
The elliptic curves in class 137904cv have rank \(2\).
Complex multiplication
The elliptic curves in class 137904cv do not have complex multiplication.Modular form 137904.2.a.cv
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.