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SageMath
E = EllipticCurve("ec1")
E.isogeny_class()
Elliptic curves in class 355740ec
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
355740.ec2 | 355740ec1 | \([0, 1, 0, -75665, -7820100]\) | \(4927700992/151875\) | \(1476578377890000\) | \([2]\) | \(2073600\) | \(1.6863\) | \(\Gamma_0(N)\)-optimal |
355740.ec1 | 355740ec2 | \([0, 1, 0, -181540, 18775700]\) | \(4253563312/1476225\) | \(229637469329452800\) | \([2]\) | \(4147200\) | \(2.0329\) |
Rank
sage: E.rank()
The elliptic curves in class 355740ec have rank \(0\).
Complex multiplication
The elliptic curves in class 355740ec do not have complex multiplication.Modular form 355740.2.a.ec
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.