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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 10350.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
10350.a1 | 10350u2 | \([1, -1, 0, -117067617, 493297644541]\) | \(-24923353462910020825/341398360424448\) | \(-2430462937006080000000000\) | \([]\) | \(2695680\) | \(3.4859\) | |
10350.a2 | 10350u1 | \([1, -1, 0, 5191758, 3404328916]\) | \(2173899265153175/1961845235712\) | \(-13966652117520000000000\) | \([]\) | \(898560\) | \(2.9366\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 10350.a have rank \(1\).
Complex multiplication
The elliptic curves in class 10350.a do not have complex multiplication.Modular form 10350.2.a.a
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.