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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 306.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 306.b1 | 306c5 | \([1, -1, 0, -249696, 48087270]\) | \(2361739090258884097/5202\) | \(3792258\) | \([2]\) | \(1024\) | \(1.3964\) | |
| 306.b2 | 306c3 | \([1, -1, 0, -15606, 754272]\) | \(576615941610337/27060804\) | \(19727326116\) | \([2, 2]\) | \(512\) | \(1.0498\) | |
| 306.b3 | 306c6 | \([1, -1, 0, -14796, 835434]\) | \(-491411892194497/125563633938\) | \(-91535889140802\) | \([2]\) | \(1024\) | \(1.3964\) | |
| 306.b4 | 306c2 | \([1, -1, 0, -1026, 10692]\) | \(163936758817/30338064\) | \(22116448656\) | \([2, 2]\) | \(256\) | \(0.70324\) | |
| 306.b5 | 306c1 | \([1, -1, 0, -306, -1836]\) | \(4354703137/352512\) | \(256981248\) | \([2]\) | \(128\) | \(0.35666\) | \(\Gamma_0(N)\)-optimal |
| 306.b6 | 306c4 | \([1, -1, 0, 2034, 60264]\) | \(1276229915423/2927177028\) | \(-2133912053412\) | \([2]\) | \(512\) | \(1.0498\) |
Rank
sage: E.rank()
The elliptic curves in class 306.b have rank \(0\).
Complex multiplication
The elliptic curves in class 306.b do not have complex multiplication.Modular form 306.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
