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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 41070.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
41070.k1 | 41070k2 | \([1, 0, 1, -205379, -35410498]\) | \(511189448451769/7077888000\) | \(13265101651968000\) | \([]\) | \(598752\) | \(1.8992\) | |
41070.k2 | 41070k1 | \([1, 0, 1, -20564, 1108946]\) | \(513108539209/12597120\) | \(23609031016320\) | \([3]\) | \(199584\) | \(1.3499\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 41070.k have rank \(0\).
Complex multiplication
The elliptic curves in class 41070.k do not have complex multiplication.Modular form 41070.2.a.k
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.