Properties

Label 25350.d
Number of curves $8$
Conductor $25350$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("25350.d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 25350.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
25350.d1 25350i7 [1, 1, 0, -22534125, -41182072875] [2] 1327104  
25350.d2 25350i8 [1, 1, 0, -1916125, -139746875] [2] 1327104  
25350.d3 25350i6 [1, 1, 0, -1409125, -643197875] [2, 2] 663552  
25350.d4 25350i5 [1, 1, 0, -1219000, 517515250] [2] 442368  
25350.d5 25350i4 [1, 1, 0, -289500, -51761250] [2] 442368  
25350.d6 25350i2 [1, 1, 0, -78250, 7600000] [2, 2] 221184  
25350.d7 25350i3 [1, 1, 0, -57125, -17221875] [2] 331776  
25350.d8 25350i1 [1, 1, 0, 6250, 586500] [2] 110592 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 25350.d have rank \(1\).

Modular form 25350.2.a.d

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{3} + q^{4} + q^{6} - 4q^{7} - q^{8} + q^{9} - q^{12} + 4q^{14} + q^{16} - 6q^{17} - q^{18} + 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.