Properties

Label 75106.a
Number of curves 4
Conductor 75106
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("75106.a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 75106.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
75106.a1 75106b4 [1, 0, 0, -249663, 33159175] [2] 1271808  
75106.a2 75106b3 [1, 0, 0, -227573, 41761021] [2] 635904  
75106.a3 75106b2 [1, 0, 0, -95033, -11281487] [2] 423936  
75106.a4 75106b1 [1, 0, 0, -6673, -130455] [2] 211968 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 75106.a have rank \(1\).

Modular form 75106.2.a.a

sage: E.q_eigenform(10)
 
\( q + q^{2} - 2q^{3} + q^{4} - 2q^{6} - 4q^{7} + q^{8} + q^{9} - 6q^{11} - 2q^{12} - 2q^{13} - 4q^{14} + q^{16} - q^{17} + q^{18} + 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.