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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 163800.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
163800.d1 | 163800dv1 | \([0, 0, 0, -107175, -12491750]\) | \(46689225424/3901625\) | \(11377138500000000\) | \([2]\) | \(1327104\) | \(1.8228\) | \(\Gamma_0(N)\)-optimal |
163800.d2 | 163800dv2 | \([0, 0, 0, 113325, -57253250]\) | \(13799183324/129390625\) | \(-1509212250000000000\) | \([2]\) | \(2654208\) | \(2.1694\) |
Rank
sage: E.rank()
The elliptic curves in class 163800.d have rank \(1\).
Complex multiplication
The elliptic curves in class 163800.d do not have complex multiplication.Modular form 163800.2.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 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.