Properties

Label 288.d
Number of curves $4$
Conductor $288$
CM \(\Q(\sqrt{-1}) \)
Rank $0$
Graph

Related objects

Downloads

Learn more about

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

Elliptic curves in class 288.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality CM discriminant
288.d1 288d2 [0, 0, 0, -99, -378] [2] 32   -16
288.d2 288d3 [0, 0, 0, -99, 378] [2] 32   -16
288.d3 288d1 [0, 0, 0, -9, 0] [2, 2] 16 \(\Gamma_0(N)\)-optimal -4
288.d4 288d4 [0, 0, 0, 36, 0] [2] 32   -4

Rank

sage: E.rank()
 

The elliptic curves in class 288.d have rank \(0\).

Complex multiplication

Each elliptic curve in class 288.d has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).

Modular form 288.2.a.d

sage: E.q_eigenform(10)
 
\( q + 2q^{5} + 6q^{13} - 2q^{17} + O(q^{20}) \)

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 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.