Properties

Label 1225.e
Number of curves $3$
Conductor $1225$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1225.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1225.e1 1225a3 \([0, 1, 1, -160883, 25929019]\) \(-250523582464/13671875\) \(-25132537841796875\) \([]\) \(6912\) \(1.9051\)  
1225.e2 1225a1 \([0, 1, 1, -1633, -28731]\) \(-262144/35\) \(-64339296875\) \([]\) \(768\) \(0.80652\) \(\Gamma_0(N)\)-optimal
1225.e3 1225a2 \([0, 1, 1, 10617, 75394]\) \(71991296/42875\) \(-78815638671875\) \([]\) \(2304\) \(1.3558\)  

Rank

sage: E.rank()
 

The elliptic curves in class 1225.e have rank \(0\).

Complex multiplication

The elliptic curves in class 1225.e do not have complex multiplication.

Modular form 1225.2.a.e

sage: E.q_eigenform(10)
 
\(q + q^{3} - 2q^{4} - 2q^{9} - 3q^{11} - 2q^{12} + 5q^{13} + 4q^{16} + 3q^{17} - 2q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.