Properties

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

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Show commands: SageMath
sage: E = EllipticCurve("d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 585.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
585.d1 585a2 \([1, -1, 1, -3593, 83782]\) \(260549802603/4225\) \(83160675\) \([2]\) \(384\) \(0.65208\)  
585.d2 585a1 \([1, -1, 1, -218, 1432]\) \(-57960603/8125\) \(-159924375\) \([2]\) \(192\) \(0.30551\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 585.2.a.d

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