Properties

Label 585.i
Number of curves $2$
Conductor $585$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("i1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 585.i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
585.i1 585c2 \([1, -1, 0, -399, -2970]\) \(260549802603/4225\) \(114075\) \([2]\) \(128\) \(0.10278\)  
585.i2 585c1 \([1, -1, 0, -24, -45]\) \(-57960603/8125\) \(-219375\) \([2]\) \(64\) \(-0.24380\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 585.i have rank \(0\).

Complex multiplication

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

Modular form 585.2.a.i

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.