Show commands:
SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 525.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
525.d1 | 525b5 | \([1, 1, 0, -19600, -1064375]\) | \(53297461115137/147\) | \(2296875\) | \([2]\) | \(512\) | \(0.87892\) | |
525.d2 | 525b3 | \([1, 1, 0, -1225, -17000]\) | \(13027640977/21609\) | \(337640625\) | \([2, 2]\) | \(256\) | \(0.53235\) | |
525.d3 | 525b4 | \([1, 1, 0, -975, 11250]\) | \(6570725617/45927\) | \(717609375\) | \([2]\) | \(256\) | \(0.53235\) | |
525.d4 | 525b6 | \([1, 1, 0, -850, -27125]\) | \(-4354703137/17294403\) | \(-270225046875\) | \([2]\) | \(512\) | \(0.87892\) | |
525.d5 | 525b2 | \([1, 1, 0, -100, -125]\) | \(7189057/3969\) | \(62015625\) | \([2, 2]\) | \(128\) | \(0.18578\) | |
525.d6 | 525b1 | \([1, 1, 0, 25, 0]\) | \(103823/63\) | \(-984375\) | \([2]\) | \(64\) | \(-0.16080\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 525.d have rank \(0\).
Complex multiplication
The elliptic curves in class 525.d do not have complex multiplication.Modular form 525.2.a.d
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.