Properties

Label 800.e
Number of curves $2$
Conductor $800$
CM \(\Q(\sqrt{-1}) \)
Rank $1$
Graph

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

Elliptic curves in class 800.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality CM discriminant
800.e1 800h1 [0, 0, 0, -5, 0] [2] 32 \(\Gamma_0(N)\)-optimal -4
800.e2 800h2 [0, 0, 0, 20, 0] [2] 64   -4

Rank

sage: E.rank()
 

The elliptic curves in class 800.e have rank \(1\).

Complex multiplication

Each elliptic curve in class 800.e has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).

Modular form 800.2.a.e

sage: E.q_eigenform(10)
 
\( q - 3q^{9} - 4q^{13} - 8q^{17} + O(q^{20}) \)

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.