Properties

Label 800a
Number of curves $4$
Conductor $800$
CM \(\Q(\sqrt{-1}) \)
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 800a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality CM discriminant
800.d3 800a1 [0, 0, 0, -25, 0] [2, 2] 64 \(\Gamma_0(N)\)-optimal -4
800.d1 800a2 [0, 0, 0, -275, -1750] [2] 128   -16
800.d2 800a3 [0, 0, 0, -275, 1750] [2] 128   -16
800.d4 800a4 [0, 0, 0, 100, 0] [2] 128   -4

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 800.2.a.a

sage: E.q_eigenform(10)
 
\( q - 3q^{9} - 6q^{13} - 2q^{17} + O(q^{20}) \)

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 Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.