Properties

Label 45675k
Number of curves $6$
Conductor $45675$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("45675.e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 45675k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
45675.e5 45675k1 [1, -1, 1, -176405, -30489028] [2] 393216 \(\Gamma_0(N)\)-optimal
45675.e4 45675k2 [1, -1, 1, -2877530, -1878058528] [2, 2] 786432  
45675.e3 45675k3 [1, -1, 1, -2932655, -1802316778] [2, 2] 1572864  
45675.e1 45675k4 [1, -1, 1, -46040405, -120230661778] [2] 1572864  
45675.e6 45675k5 [1, -1, 1, 2808220, -8002461778] [2] 3145728  
45675.e2 45675k6 [1, -1, 1, -9555530, 9244638722] [2] 3145728  

Rank

sage: E.rank()
 

The elliptic curves in class 45675k have rank \(0\).

Modular form 45675.2.a.e

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.