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Properties

Label 5225a
Number of curves $2$
Conductor $5225$
CM no
Rank $1$
Graph

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

Elliptic curves in class 5225a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5225.c2 5225a1 \([1, 1, 0, -150, -625]\) \(24137569/5225\) \(81640625\) \([2]\) \(1536\) \(0.23210\) \(\Gamma_0(N)\)-optimal
5225.c1 5225a2 \([1, 1, 0, -775, 7500]\) \(3301293169/218405\) \(3412578125\) \([2]\) \(3072\) \(0.57868\)  

Rank

sage: E.rank()
 

The elliptic curves in class 5225a have rank \(1\).

Complex multiplication

The elliptic curves in class 5225a do not have complex multiplication.

Modular form 5225.2.a.a

sage: E.q_eigenform(10)
 
\(q + q^{2} + 2 q^{3} - q^{4} + 2 q^{6} + 2 q^{7} - 3 q^{8} + q^{9} - q^{11} - 2 q^{12} - 6 q^{13} + 2 q^{14} - q^{16} + q^{18} - q^{19} + O(q^{20})\) 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 Cremona 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 Cremona labels.