Properties

Label 3648.y
Number of curves $2$
Conductor $3648$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("y1")
 
E.isogeny_class()
 

Elliptic curves in class 3648.y

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3648.y1 3648be2 \([0, 1, 0, -17561, 889893]\) \(-9358714467168256/22284891\) \(-1426233024\) \([]\) \(4800\) \(0.99798\)  
3648.y2 3648be1 \([0, 1, 0, 79, 333]\) \(841232384/1121931\) \(-71803584\) \([]\) \(960\) \(0.19326\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 3648.y have rank \(1\).

Complex multiplication

The elliptic curves in class 3648.y do not have complex multiplication.

Modular form 3648.2.a.y

sage: E.q_eigenform(10)
 
\(q + q^{3} - q^{5} - 3 q^{7} + q^{9} - 3 q^{11} + 6 q^{13} - q^{15} + 3 q^{17} - q^{19} + O(q^{20})\) 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.