Properties

Label 338130.z
Number of curves $4$
Conductor $338130$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("z1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 338130.z

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
338130.z1 338130z4 [1, -1, 0, -1307490, 345008060] [2] 15925248  
338130.z2 338130z2 [1, -1, 0, -1155765, 478536175] [2] 5308416  
338130.z3 338130z1 [1, -1, 0, -72015, 7538425] [2] 2654208 \(\Gamma_0(N)\)-optimal
338130.z4 338130z3 [1, -1, 0, 253110, 38194100] [2] 7962624  

Rank

sage: E.rank()
 

The elliptic curves in class 338130.z have rank \(0\).

Complex multiplication

The elliptic curves in class 338130.z do not have complex multiplication.

Modular form 338130.2.a.z

sage: E.q_eigenform(10)
 
\( q - q^{2} + q^{4} - q^{5} + 4q^{7} - q^{8} + q^{10} + q^{13} - 4q^{14} + q^{16} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.