Properties

Label 9009.k
Number of curves $4$
Conductor $9009$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("k1")
 
E.isogeny_class()
 

Elliptic curves in class 9009.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
9009.k1 9009c4 \([1, -1, 0, -89241, 10283462]\) \(107818231938348177/4463459\) \(3253861611\) \([2]\) \(19456\) \(1.3115\)  
9009.k2 9009c3 \([1, -1, 0, -9051, -59860]\) \(112489728522417/62811265517\) \(45789412561893\) \([2]\) \(19456\) \(1.3115\)  
9009.k3 9009c2 \([1, -1, 0, -5586, 161207]\) \(26444947540257/169338169\) \(123447525201\) \([2, 2]\) \(9728\) \(0.96496\)  
9009.k4 9009c1 \([1, -1, 0, -141, 5480]\) \(-426957777/17320303\) \(-12626500887\) \([2]\) \(4864\) \(0.61839\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 9009.k have rank \(0\).

Complex multiplication

The elliptic curves in class 9009.k do not have complex multiplication.

Modular form 9009.2.a.k

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{4} + 2 q^{5} - q^{7} - 3 q^{8} + 2 q^{10} - q^{11} - q^{13} - q^{14} - q^{16} + 2 q^{17} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.