Properties

Label 800.f
Number of curves $2$
Conductor $800$
CM \(\Q(\sqrt{-1}) \)
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("f1")
 
E.isogeny_class()
 

Elliptic curves in class 800.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
800.f1 800d1 \([0, 0, 0, -125, 0]\) \(1728\) \(125000000\) \([2]\) \(160\) \(0.24312\) \(\Gamma_0(N)\)-optimal \(-4\)
800.f2 800d2 \([0, 0, 0, 500, 0]\) \(1728\) \(-8000000000\) \([2]\) \(320\) \(0.58969\)   \(-4\)

Rank

sage: E.rank()
 

The elliptic curves in class 800.f have rank \(0\).

Complex multiplication

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

Modular form 800.2.a.f

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