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Properties

Label 325.e
Number of curves $2$
Conductor $325$
CM no
Rank $0$
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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 325.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
325.e1 325d2 \([0, 1, 1, -2458, 42369]\) \(4206161920/371293\) \(145036328125\) \([]\) \(420\) \(0.88191\)  
325.e2 325d1 \([0, 1, 1, -508, -4581]\) \(23242854400/13\) \(8125\) \([]\) \(84\) \(0.077191\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 325.e have rank \(0\).

Complex multiplication

The elliptic curves in class 325.e do not have complex multiplication.

Modular form 325.2.a.e

sage: E.q_eigenform(10)
 
\(q + 2 q^{2} + q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{7} - 2 q^{9} + 2 q^{11} + 2 q^{12} - q^{13} + 4 q^{14} - 4 q^{16} + 2 q^{17} - 4 q^{18} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.