Properties

Label 23104.bf
Number of curves $4$
Conductor $23104$
CM \(\Q(\sqrt{-1}) \)
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bf1")
 
E.isogeny_class()
 

Elliptic curves in class 23104.bf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
23104.bf1 23104bo3 \([0, 0, 0, -15884, -768208]\) \(287496\) \(1541599428608\) \([2]\) \(28800\) \(1.2014\)   \(-16\)
23104.bf2 23104bo4 \([0, 0, 0, -15884, 768208]\) \(287496\) \(1541599428608\) \([2]\) \(28800\) \(1.2014\)   \(-16\)
23104.bf3 23104bo2 \([0, 0, 0, -1444, 0]\) \(1728\) \(192699928576\) \([2, 2]\) \(14400\) \(0.85483\)   \(-4\)
23104.bf4 23104bo1 \([0, 0, 0, 361, 0]\) \(1728\) \(-3010936384\) \([2]\) \(7200\) \(0.50826\) \(\Gamma_0(N)\)-optimal \(-4\)

Rank

sage: E.rank()
 

The elliptic curves in class 23104.bf have rank \(1\).

Complex multiplication

Each elliptic curve in class 23104.bf has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).

Modular form 23104.2.a.bf

sage: E.q_eigenform(10)
 
\(q + 2 q^{5} - 3 q^{9} + 6 q^{13} + 2 q^{17} + O(q^{20})\) 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.