Properties

Label 2304.d
Number of curves $2$
Conductor $2304$
CM \(\Q(\sqrt{-1}) \)
Rank $0$
Graph

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

Elliptic curves in class 2304.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
2304.d1 2304j2 \([0, 0, 0, -216, 0]\) \(1728\) \(644972544\) \([2]\) \(768\) \(0.37986\)   \(-4\)
2304.d2 2304j1 \([0, 0, 0, 54, 0]\) \(1728\) \(-10077696\) \([2]\) \(384\) \(0.033287\) \(\Gamma_0(N)\)-optimal \(-4\)

Rank

sage: E.rank()
 

The elliptic curves in class 2304.d have rank \(0\).

Complex multiplication

Each elliptic curve in class 2304.d has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).

Modular form 2304.2.a.d

sage: E.q_eigenform(10)
 
\(q - 2q^{5} + 4q^{13} - 8q^{17} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.