Properties

Label 28224.fc
Number of curves $4$
Conductor $28224$
CM \(\Q(\sqrt{-1}) \)
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("fc1")
 
E.isogeny_class()
 

Elliptic curves in class 28224.fc

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
28224.fc1 28224fp3 \([0, 0, 0, -19404, -1037232]\) \(287496\) \(2810384252928\) \([2]\) \(49152\) \(1.2514\)   \(-16\)
28224.fc2 28224fp4 \([0, 0, 0, -19404, 1037232]\) \(287496\) \(2810384252928\) \([2]\) \(49152\) \(1.2514\)   \(-16\)
28224.fc3 28224fp2 \([0, 0, 0, -1764, 0]\) \(1728\) \(351298031616\) \([2, 2]\) \(24576\) \(0.90488\)   \(-4\)
28224.fc4 28224fp1 \([0, 0, 0, 441, 0]\) \(1728\) \(-5489031744\) \([2]\) \(12288\) \(0.55830\) \(\Gamma_0(N)\)-optimal \(-4\)

Rank

sage: E.rank()
 

The elliptic curves in class 28224.fc have rank \(0\).

Complex multiplication

Each elliptic curve in class 28224.fc has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).

Modular form 28224.2.a.fc

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.