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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 42978u
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
42978.x1 | 42978u1 | \([1, 0, 0, -162, -588]\) | \(470366406433/129621648\) | \(129621648\) | \([2]\) | \(17152\) | \(0.26419\) | \(\Gamma_0(N)\)-optimal |
42978.x2 | 42978u2 | \([1, 0, 0, 418, -3720]\) | \(8075838390047/10764183924\) | \(-10764183924\) | \([2]\) | \(34304\) | \(0.61077\) |
Rank
sage: E.rank()
The elliptic curves in class 42978u have rank \(0\).
Complex multiplication
The elliptic curves in class 42978u do not have complex multiplication.Modular form 42978.2.a.u
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.