Properties

Label 7605q
Number of curves 8
Conductor 7605
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("7605.g1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 7605q

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
7605.g7 7605q1 [1, -1, 1, -32, -11046] [2] 4608 \(\Gamma_0(N)\)-optimal
7605.g6 7605q2 [1, -1, 1, -7637, -251364] [2, 2] 9216  
7605.g4 7605q3 [1, -1, 1, -121712, -16313124] [2] 18432  
7605.g5 7605q4 [1, -1, 1, -15242, 338784] [2, 2] 18432  
7605.g2 7605q5 [1, -1, 1, -205367, 35854134] [2, 2] 36864  
7605.g8 7605q6 [1, -1, 1, 53203, 2501646] [2] 36864  
7605.g1 7605q7 [1, -1, 1, -3285392, 2292896454] [2] 73728  
7605.g3 7605q8 [1, -1, 1, -167342, 49512714] [2] 73728  

Rank

sage: E.rank()
 

The elliptic curves in class 7605q have rank \(1\).

Modular form 7605.2.a.g

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{4} + q^{5} + 3q^{8} - q^{10} - 4q^{11} - q^{16} - 2q^{17} - 4q^{19} + O(q^{20}) \)

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.