Properties

Label 227430cf
Number of curves 8
Conductor 227430
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("227430.dz1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 227430cf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
227430.dz7 227430cf1 [1, -1, 1, 682222, 163262481] [2] 7077888 \(\Gamma_0(N)\)-optimal
227430.dz6 227430cf2 [1, -1, 1, -3476498, 1464110097] [2, 2] 14155776  
227430.dz4 227430cf3 [1, -1, 1, -48962498, 131845180497] [2] 28311552  
227430.dz5 227430cf4 [1, -1, 1, -24530018, -45712617519] [2, 2] 28311552  
227430.dz8 227430cf5 [1, -1, 1, 4126162, -146146797183] [2] 56623104  
227430.dz2 227430cf6 [1, -1, 1, -390042518, -2964841647519] [2, 2] 56623104  
227430.dz3 227430cf7 [1, -1, 1, -387605768, -3003716582319] [2] 113246208  
227430.dz1 227430cf8 [1, -1, 1, -6240679268, -189754610782719] [2] 113246208  

Rank

sage: E.rank()
 

The elliptic curves in class 227430cf have rank \(0\).

Modular form 227430.2.a.dz

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{4} - q^{5} - q^{7} + q^{8} - q^{10} + 4q^{11} + 2q^{13} - q^{14} + q^{16} - 2q^{17} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.