Properties

Label 43245.h
Number of curves 8
Conductor 43245
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("43245.h1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 43245.h

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
43245.h1 43245b8 [1, -1, 0, -18682020, -31075534089] [2] 983040  
43245.h2 43245b6 [1, -1, 0, -1167795, -485188704] [2, 2] 491520  
43245.h3 43245b7 [1, -1, 0, -951570, -670580019] [2] 983040  
43245.h4 43245b4 [1, -1, 0, -692100, 221789205] [2] 245760  
43245.h5 43245b3 [1, -1, 0, -86670, -4520529] [2, 2] 245760  
43245.h6 43245b2 [1, -1, 0, -43425, 3445200] [2, 2] 122880  
43245.h7 43245b1 [1, -1, 0, -180, 149931] [2] 61440 \(\Gamma_0(N)\)-optimal
43245.h8 43245b5 [1, -1, 0, 302535, -34177950] [2] 491520  

Rank

sage: E.rank()
 

The elliptic curves in class 43245.h have rank \(1\).

Modular form 43245.2.a.h

sage: E.q_eigenform(10)
 
\( q + q^{2} - q^{4} - q^{5} - 3q^{8} - q^{10} - 4q^{11} + 2q^{13} - q^{16} + 2q^{17} + 4q^{19} + O(q^{20}) \)

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.