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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 32340u
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
32340.r2 | 32340u1 | \([0, -1, 0, -485, 3585]\) | \(1007878144/179685\) | \(2253968640\) | \([]\) | \(15552\) | \(0.51398\) | \(\Gamma_0(N)\)-optimal |
32340.r1 | 32340u2 | \([0, -1, 0, -37445, 2801457]\) | \(462893166690304/4125\) | \(51744000\) | \([]\) | \(46656\) | \(1.0633\) |
Rank
sage: E.rank()
The elliptic curves in class 32340u have rank \(1\).
Complex multiplication
The elliptic curves in class 32340u do not have complex multiplication.Modular form 32340.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.