Properties

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

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Show commands: SageMath
sage: E = EllipticCurve("e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 585.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
585.e1 585b2 \([0, 0, 1, -378, -2842]\) \(-303464448/1625\) \(-31984875\) \([]\) \(144\) \(0.28459\)  
585.e2 585b1 \([0, 0, 1, 12, -21]\) \(7077888/10985\) \(-296595\) \([3]\) \(48\) \(-0.26471\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 585.2.a.e

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

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.