Properties

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

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Show commands: SageMath
sage: E = EllipticCurve("fm1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 28224.fm

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
28224.fm1 28224ft4 \([0, 0, 0, -37933644, -89925934448]\) \(268498407453697/252\) \(5665734653902848\) \([2]\) \(1179648\) \(2.7511\)  
28224.fm2 28224ft6 \([0, 0, 0, -25797324, 49948258192]\) \(84448510979617/933897762\) \(20996892513356009177088\) \([2]\) \(2359296\) \(3.0977\)  
28224.fm3 28224ft3 \([0, 0, 0, -2935884, -685259120]\) \(124475734657/63011844\) \(1416699953004445433856\) \([2, 2]\) \(1179648\) \(2.7511\)  
28224.fm4 28224ft2 \([0, 0, 0, -2371404, -1404406640]\) \(65597103937/63504\) \(1427765132783517696\) \([2, 2]\) \(589824\) \(2.4045\)  
28224.fm5 28224ft1 \([0, 0, 0, -113484, -32494448]\) \(-7189057/16128\) \(-362607017849782272\) \([2]\) \(294912\) \(2.0579\) \(\Gamma_0(N)\)-optimal
28224.fm6 28224ft5 \([0, 0, 0, 10893876, -5293335152]\) \(6359387729183/4218578658\) \(-94846616241450678484992\) \([2]\) \(2359296\) \(3.0977\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 28224.2.a.fm

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.