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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 7056.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
7056.p1 | 7056bx5 | \([0, 0, 0, -5532051, 5008150546]\) | \(53297461115137/147\) | \(51640810647552\) | \([2]\) | \(98304\) | \(2.2896\) | |
7056.p2 | 7056bx4 | \([0, 0, 0, -345891, 78186850]\) | \(13027640977/21609\) | \(7591199165190144\) | \([2, 2]\) | \(49152\) | \(1.9430\) | |
7056.p3 | 7056bx3 | \([0, 0, 0, -275331, -55270334]\) | \(6570725617/45927\) | \(16134064698028032\) | \([2]\) | \(49152\) | \(1.9430\) | |
7056.p4 | 7056bx6 | \([0, 0, 0, -240051, 126936754]\) | \(-4354703137/17294403\) | \(-6075489731873845248\) | \([2]\) | \(98304\) | \(2.2896\) | |
7056.p5 | 7056bx2 | \([0, 0, 0, -28371, 394450]\) | \(7189057/3969\) | \(1394301887483904\) | \([2, 2]\) | \(24576\) | \(1.5965\) | |
7056.p6 | 7056bx1 | \([0, 0, 0, 6909, 48706]\) | \(103823/63\) | \(-22131775991808\) | \([2]\) | \(12288\) | \(1.2499\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 7056.p have rank \(0\).
Complex multiplication
The elliptic curves in class 7056.p do not have complex multiplication.Modular form 7056.2.a.p
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 8 & 4 & 4 & 8 \\ 2 & 1 & 4 & 2 & 2 & 4 \\ 8 & 4 & 1 & 8 & 2 & 4 \\ 4 & 2 & 8 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.