Properties

Label 525.d
Number of curves $6$
Conductor $525$
CM no
Rank $0$
Graph

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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)
 
\(q + q^{2} - q^{3} - q^{4} - q^{6} + q^{7} - 3 q^{8} + q^{9} + 4 q^{11} + q^{12} + 2 q^{13} + q^{14} - q^{16} + 6 q^{17} + q^{18} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

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.