Properties

Label 585.h
Number of curves $2$
Conductor $585$
CM no
Rank $1$
Graph

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

Elliptic curves in class 585.h

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
585.h1 585h1 \([1, -1, 0, -9, 0]\) \(117649/65\) \(47385\) \([2]\) \(48\) \(-0.41231\) \(\Gamma_0(N)\)-optimal
585.h2 585h2 \([1, -1, 0, 36, -27]\) \(6967871/4225\) \(-3080025\) \([2]\) \(96\) \(-0.065735\)  

Rank

sage: E.rank()
 

The elliptic curves in class 585.h have rank \(1\).

Complex multiplication

The elliptic curves in class 585.h do not have complex multiplication.

Modular form 585.2.a.h

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.