Properties

Label 425880.eb
Number of curves $6$
Conductor $425880$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("425880.eb1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 425880.eb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
425880.eb1 425880eb5 [0, 0, 0, -15940587, 24495983174] [2] 18874368 \(\Gamma_0(N)\)-optimal*
425880.eb2 425880eb3 [0, 0, 0, -1034787, 351568334] [2, 2] 9437184 \(\Gamma_0(N)\)-optimal*
425880.eb3 425880eb2 [0, 0, 0, -274287, -49823566] [2, 2] 4718592 \(\Gamma_0(N)\)-optimal*
425880.eb4 425880eb1 [0, 0, 0, -266682, -53007019] [2] 2359296 \(\Gamma_0(N)\)-optimal*
425880.eb5 425880eb4 [0, 0, 0, 364533, -247474474] [2] 9437184  
425880.eb6 425880eb6 [0, 0, 0, 1703013, 1896235094] [2] 18874368  
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 425880.eb4.

Rank

sage: E.rank()
 

The elliptic curves in class 425880.eb have rank \(1\).

Modular form 425880.2.a.eb

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

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 LMFDB numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.