Properties

Label 6045.h
Number of curves $4$
Conductor $6045$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("h1")
 
E.isogeny_class()
 

Elliptic curves in class 6045.h

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6045.h1 6045b3 \([1, 1, 0, -2483, -48672]\) \(1694053550246329/13280865\) \(13280865\) \([2]\) \(4608\) \(0.53992\)  
6045.h2 6045b2 \([1, 1, 0, -158, -777]\) \(440537367529/36542025\) \(36542025\) \([2, 2]\) \(2304\) \(0.19335\)  
6045.h3 6045b1 \([1, 1, 0, -33, 48]\) \(4165509529/755625\) \(755625\) \([2]\) \(1152\) \(-0.15323\) \(\Gamma_0(N)\)-optimal
6045.h4 6045b4 \([1, 1, 0, 167, -3182]\) \(510273943271/4862338065\) \(-4862338065\) \([2]\) \(4608\) \(0.53992\)  

Rank

sage: E.rank()
 

The elliptic curves in class 6045.h have rank \(0\).

Complex multiplication

The elliptic curves in class 6045.h do not have complex multiplication.

Modular form 6045.2.a.h

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} - q^{4} - q^{5} - q^{6} + 4 q^{7} - 3 q^{8} + q^{9} - q^{10} - 4 q^{11} + q^{12} - q^{13} + 4 q^{14} + q^{15} - q^{16} - 6 q^{17} + q^{18} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.