Properties

Label 28224.es
Number of curves $6$
Conductor $28224$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("es1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 28224.es

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
28224.es1 28224fw6 \([0, 0, 0, -22128204, 40065204368]\) \(53297461115137/147\) \(3305011881443328\) \([2]\) \(786432\) \(2.6362\)  
28224.es2 28224fw4 \([0, 0, 0, -1383564, 625494800]\) \(13027640977/21609\) \(485836746572169216\) \([2, 2]\) \(393216\) \(2.2896\)  
28224.es3 28224fw3 \([0, 0, 0, -1101324, -442162672]\) \(6570725617/45927\) \(1032580140673794048\) \([2]\) \(393216\) \(2.2896\)  
28224.es4 28224fw5 \([0, 0, 0, -960204, 1015494032]\) \(-4354703137/17294403\) \(-388831342839926095872\) \([2]\) \(786432\) \(2.6362\)  
28224.es5 28224fw2 \([0, 0, 0, -113484, 3155600]\) \(7189057/3969\) \(89235320798969856\) \([2, 2]\) \(196608\) \(1.9430\)  
28224.es6 28224fw1 \([0, 0, 0, 27636, 389648]\) \(103823/63\) \(-1416433663475712\) \([2]\) \(98304\) \(1.5965\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 28224.2.a.es

sage: E.q_eigenform(10)
 
\(q + 2q^{5} - 4q^{11} - 2q^{13} - 6q^{17} - 4q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.