Properties

Label 225.a
Number of curves $2$
Conductor $225$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

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

Elliptic curves in class 225.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
225.a1 225e1 \([0, 0, 1, -75, 256]\) \(-102400/3\) \(-1366875\) \([]\) \(48\) \(-0.043792\) \(\Gamma_0(N)\)-optimal
225.a2 225e2 \([0, 0, 1, 375, -12344]\) \(20480/243\) \(-69198046875\) \([]\) \(240\) \(0.76093\)  

Rank

sage: E.rank()
 

The elliptic curves in class 225.a have rank \(1\).

Complex multiplication

The elliptic curves in class 225.a do not have complex multiplication.

Modular form 225.2.a.a

sage: E.q_eigenform(10)
 
\(q - 2 q^{2} + 2 q^{4} - 3 q^{7} - 2 q^{11} + q^{13} + 6 q^{14} - 4 q^{16} - 2 q^{17} - 5 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.