Properties

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

Related objects

Downloads

Learn more

Show commands for: SageMath
sage: E = EllipticCurve("d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 800.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
800.d1 800a2 \([0, 0, 0, -275, -1750]\) \(287496\) \(8000000\) \([2]\) \(128\) \(0.18733\)   \(-16\)
800.d2 800a3 \([0, 0, 0, -275, 1750]\) \(287496\) \(8000000\) \([2]\) \(128\) \(0.18733\)   \(-16\)
800.d3 800a1 \([0, 0, 0, -25, 0]\) \(1728\) \(1000000\) \([2, 2]\) \(64\) \(-0.15924\) \(\Gamma_0(N)\)-optimal \(-4\)
800.d4 800a4 \([0, 0, 0, 100, 0]\) \(1728\) \(-64000000\) \([2]\) \(128\) \(0.18733\)   \(-4\)

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 800.2.a.d

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.