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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 32368w
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
32368.r3 | 32368w1 | \([0, 0, 0, -86411, -9619654]\) | \(721734273/13328\) | \(1317705808412672\) | \([2]\) | \(110592\) | \(1.6957\) | \(\Gamma_0(N)\)-optimal |
32368.r2 | 32368w2 | \([0, 0, 0, -178891, 14591610]\) | \(6403769793/2775556\) | \(274412234601938944\) | \([2, 2]\) | \(221184\) | \(2.0423\) | |
32368.r4 | 32368w3 | \([0, 0, 0, 607189, 108135130]\) | \(250404380127/196003234\) | \(-19378346331742806016\) | \([4]\) | \(442368\) | \(2.3889\) | |
32368.r1 | 32368w4 | \([0, 0, 0, -2444651, 1470568986]\) | \(16342588257633/8185058\) | \(809236079591432192\) | \([2]\) | \(442368\) | \(2.3889\) |
Rank
sage: E.rank()
The elliptic curves in class 32368w have rank \(1\).
Complex multiplication
The elliptic curves in class 32368w do not have complex multiplication.Modular form 32368.2.a.w
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.