Properties

Label 1525.a
Number of curves $2$
Conductor $1525$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 1525.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1525.a1 1525c2 \([1, 0, 0, -33, -68]\) \(31855013/3721\) \(465125\) \([2]\) \(224\) \(-0.18087\)  
1525.a2 1525c1 \([1, 0, 0, -8, 7]\) \(456533/61\) \(7625\) \([2]\) \(112\) \(-0.52745\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 1525.a have rank \(2\).

Complex multiplication

The elliptic curves in class 1525.a do not have complex multiplication.

Modular form 1525.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{2} - 2 q^{3} - q^{4} + 2 q^{6} - 4 q^{7} + 3 q^{8} + q^{9} + 2 q^{12} - 4 q^{13} + 4 q^{14} - q^{16} - 6 q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.