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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 3234.u
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3234.u1 | 3234v2 | \([1, 0, 0, -52459, 4620209]\) | \(46546832455691959/748268928\) | \(256656242304\) | \([2]\) | \(10752\) | \(1.3221\) | |
3234.u2 | 3234v1 | \([1, 0, 0, -3179, 76593]\) | \(-10358806345399/1445216256\) | \(-495709175808\) | \([2]\) | \(5376\) | \(0.97555\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3234.u have rank \(1\).
Complex multiplication
The elliptic curves in class 3234.u do not have complex multiplication.Modular form 3234.2.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 LMFDB 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 LMFDB labels.