Properties

Label 28224.fl
Number of curves $6$
Conductor $28224$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("fl1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 28224.fl

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
28224.fl1 28224by6 \([0, 0, 0, -677964, -214860688]\) \(3065617154/9\) \(101173833105408\) \([2]\) \(196608\) \(1.9166\)  
28224.fl2 28224by4 \([0, 0, 0, -113484, 14713328]\) \(28756228/3\) \(16862305517568\) \([2]\) \(98304\) \(1.5701\)  
28224.fl3 28224by3 \([0, 0, 0, -42924, -3265360]\) \(1556068/81\) \(455282248974336\) \([2, 2]\) \(98304\) \(1.5701\)  
28224.fl4 28224by2 \([0, 0, 0, -7644, 192080]\) \(35152/9\) \(12646729138176\) \([2, 2]\) \(49152\) \(1.2235\)  
28224.fl5 28224by1 \([0, 0, 0, 1176, 19208]\) \(2048/3\) \(-263473523712\) \([2]\) \(24576\) \(0.87691\) \(\Gamma_0(N)\)-optimal
28224.fl6 28224by5 \([0, 0, 0, 27636, -12946192]\) \(207646/6561\) \(-73755724333842432\) \([2]\) \(196608\) \(1.9166\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 28224.2.a.fl

sage: E.q_eigenform(10)
 
\(q + 2q^{5} + 4q^{11} - 2q^{13} + 2q^{17} - 4q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.