Properties

Label 215475.t
Number of curves $6$
Conductor $215475$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("215475.t1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 215475.t

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
215475.t1 215475p6 [1, 0, 0, -85235238, -302844873033] [2] 22020096  
215475.t2 215475p4 [1, 0, 0, -5868613, -3712063408] [2, 2] 11010048  
215475.t3 215475p2 [1, 0, 0, -2298488, 1296821967] [2, 2] 5505024  
215475.t4 215475p1 [1, 0, 0, -2277363, 1322615592] [2] 2752512 \(\Gamma_0(N)\)-optimal
215475.t5 215475p3 [1, 0, 0, 933637, 4654999842] [2] 11010048  
215475.t6 215475p5 [1, 0, 0, 16376012, -25133637283] [2] 22020096  

Rank

sage: E.rank()
 

The elliptic curves in class 215475.t have rank \(0\).

Modular form 215475.2.a.t

sage: E.q_eigenform(10)
 
\( q - q^{2} + q^{3} - q^{4} - q^{6} + 3q^{8} + q^{9} - 4q^{11} - q^{12} - q^{16} - q^{17} - q^{18} + 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.