Properties

Label 28224.fr
Number of curves $2$
Conductor $28224$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("fr1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 28224.fr

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
28224.fr1 28224e1 \([0, 0, 0, -292236, 60812528]\) \(-67645179/8\) \(-326420926562304\) \([]\) \(193536\) \(1.8105\) \(\Gamma_0(N)\)-optimal
28224.fr2 28224e2 \([0, 0, 0, 37044, 187738992]\) \(189/512\) \(-15229494749690855424\) \([]\) \(580608\) \(2.3598\)  

Rank

sage: E.rank()
 

The elliptic curves in class 28224.fr have rank \(1\).

Complex multiplication

The elliptic curves in class 28224.fr do not have complex multiplication.

Modular form 28224.2.a.fr

sage: E.q_eigenform(10)
 
\(q + 3q^{5} - 3q^{11} - 2q^{13} - 6q^{17} - 2q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.