Properties

Label 288e
Number of curves $2$
Conductor $288$
CM \(\Q(\sqrt{-1}) \)
Rank $0$
Graph

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

Elliptic curves in class 288e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality CM discriminant
288.e2 288e1 [0, 0, 0, 27, 0] [2] 48 \(\Gamma_0(N)\)-optimal -4
288.e1 288e2 [0, 0, 0, -108, 0] [2] 96   -4

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 288.2.a.e

sage: E.q_eigenform(10)
 
\( q + 4q^{5} - 6q^{13} + 8q^{17} + O(q^{20}) \)

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 Cremona 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 Cremona labels.