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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 576.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 576.b1 | 576i5 | \([0, 0, 0, -13836, -626416]\) | \(3065617154/9\) | \(859963392\) | \([2]\) | \(512\) | \(0.94368\) | |
| 576.b2 | 576i4 | \([0, 0, 0, -2316, 42896]\) | \(28756228/3\) | \(143327232\) | \([2]\) | \(256\) | \(0.59710\) | |
| 576.b3 | 576i3 | \([0, 0, 0, -876, -9520]\) | \(1556068/81\) | \(3869835264\) | \([2, 2]\) | \(256\) | \(0.59710\) | |
| 576.b4 | 576i2 | \([0, 0, 0, -156, 560]\) | \(35152/9\) | \(107495424\) | \([2, 2]\) | \(128\) | \(0.25053\) | |
| 576.b5 | 576i1 | \([0, 0, 0, 24, 56]\) | \(2048/3\) | \(-2239488\) | \([2]\) | \(64\) | \(-0.096046\) | \(\Gamma_0(N)\)-optimal |
| 576.b6 | 576i6 | \([0, 0, 0, 564, -37744]\) | \(207646/6561\) | \(-626913312768\) | \([2]\) | \(512\) | \(0.94368\) |
Rank
sage: E.rank()
The elliptic curves in class 576.b have rank \(1\).
Complex multiplication
The elliptic curves in class 576.b do not have complex multiplication.Modular form 576.2.a.b
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
