Properties

Label 27075.f
Number of curves 8
Conductor 27075
CM no
Rank 2
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("27075.f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 27075.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
27075.f1 27075g8 [1, 1, 1, -19494188, 33120687656] [2] 663552  
27075.f2 27075g6 [1, 1, 1, -1218563, 516972656] [2, 2] 331776  
27075.f3 27075g7 [1, 1, 1, -992938, 714620156] [2] 663552  
27075.f4 27075g4 [1, 1, 1, -722188, -236524594] [2] 165888  
27075.f5 27075g3 [1, 1, 1, -90438, 4803906] [2, 2] 165888  
27075.f6 27075g2 [1, 1, 1, -45313, -3679594] [2, 2] 82944  
27075.f7 27075g1 [1, 1, 1, -188, -159844] [2] 41472 \(\Gamma_0(N)\)-optimal
27075.f8 27075g5 [1, 1, 1, 315687, 36481656] [2] 331776  

Rank

sage: E.rank()
 

The elliptic curves in class 27075.f have rank \(2\).

Modular form 27075.2.a.f

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{3} - q^{4} + q^{6} + 3q^{8} + q^{9} - 4q^{11} + q^{12} - 2q^{13} - q^{16} - 2q^{17} - q^{18} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.