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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 6690.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6690.i1 | 6690i2 | \([1, 0, 0, -1751, -31269]\) | \(-593741837169649/74854577250\) | \(-74854577250\) | \([]\) | \(8856\) | \(0.82062\) | |
6690.i2 | 6690i1 | \([1, 0, 0, 139, 105]\) | \(296874449711/175572360\) | \(-175572360\) | \([3]\) | \(2952\) | \(0.27132\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6690.i have rank \(0\).
Complex multiplication
The elliptic curves in class 6690.i do not have complex multiplication.Modular form 6690.2.a.i
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.