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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 63525.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
63525.j1 | 63525i6 | \([1, 1, 1, -2371663, 1404824906]\) | \(53297461115137/147\) | \(4069054171875\) | \([2]\) | \(655360\) | \(2.0779\) | |
63525.j2 | 63525i4 | \([1, 1, 1, -148288, 21885656]\) | \(13027640977/21609\) | \(598150963265625\) | \([2, 2]\) | \(327680\) | \(1.7313\) | |
63525.j3 | 63525i3 | \([1, 1, 1, -118038, -15563844]\) | \(6570725617/45927\) | \(1271288781984375\) | \([2]\) | \(327680\) | \(1.7313\) | |
63525.j4 | 63525i5 | \([1, 1, 1, -102913, 35588906]\) | \(-4354703137/17294403\) | \(-478720154266921875\) | \([2]\) | \(655360\) | \(2.0779\) | |
63525.j5 | 63525i2 | \([1, 1, 1, -12163, 105656]\) | \(7189057/3969\) | \(109864462640625\) | \([2, 2]\) | \(163840\) | \(1.3847\) | |
63525.j6 | 63525i1 | \([1, 1, 1, 2962, 14906]\) | \(103823/63\) | \(-1743880359375\) | \([2]\) | \(81920\) | \(1.0382\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 63525.j have rank \(2\).
Complex multiplication
The elliptic curves in class 63525.j do not have complex multiplication.Modular form 63525.2.a.j
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.