Properties

Label 704k
Number of curves 3
Conductor 704
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("704.c1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 704k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
704.c3 704k1 [0, -1, 0, -1, -1] [] 16 \(\Gamma_0(N)\)-optimal
704.c2 704k2 [0, -1, 0, -41, 199] [] 80  
704.c1 704k3 [0, -1, 0, -31281, 2139919] [] 400  

Rank

sage: E.rank()
 

The elliptic curves in class 704k have rank \(1\).

Modular form 704.2.a.c

sage: E.q_eigenform(10)
 
\( q - q^{3} - q^{5} + 2q^{7} - 2q^{9} + q^{11} - 4q^{13} + q^{15} - 2q^{17} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrr} 1 & 5 & 25 \\ 5 & 1 & 5 \\ 25 & 5 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.