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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 351329d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
351329.d1 | 351329d1 | \([0, -1, 1, -45947, 4448595]\) | \(-2258403328/480491\) | \(-2282382336862331\) | \([]\) | \(1693440\) | \(1.6684\) | \(\Gamma_0(N)\)-optimal |
351329.d2 | 351329d2 | \([0, -1, 1, 323873, -25747208]\) | \(790939860992/517504691\) | \(-2458201227456494531\) | \([]\) | \(5080320\) | \(2.2177\) |
Rank
sage: E.rank()
The elliptic curves in class 351329d have rank \(1\).
Complex multiplication
The elliptic curves in class 351329d do not have complex multiplication.Modular form 351329.2.a.d
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.