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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 433200.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
433200.k1 | 433200k3 | \([0, -1, 0, -61806208, -187001401088]\) | \(8671983378625/82308\) | \(247824151894272000000\) | \([2]\) | \(44789760\) | \(3.0756\) | |
433200.k2 | 433200k4 | \([0, -1, 0, -60362208, -196156361088]\) | \(-8078253774625/846825858\) | \(-2549738786764217472000000\) | \([2]\) | \(89579520\) | \(3.4222\) | |
433200.k3 | 433200k1 | \([0, -1, 0, -1158208, 37030912]\) | \(57066625/32832\) | \(98855063359488000000\) | \([2]\) | \(14929920\) | \(2.5263\) | \(\Gamma_0(N)\)-optimal* |
433200.k4 | 433200k2 | \([0, -1, 0, 4617792, 291174912]\) | \(3616805375/2105352\) | \(-6339080937927168000000\) | \([2]\) | \(29859840\) | \(2.8729\) |
Rank
sage: E.rank()
The elliptic curves in class 433200.k have rank \(1\).
Complex multiplication
The elliptic curves in class 433200.k do not have complex multiplication.Modular form 433200.2.a.k
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.