Properties

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

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Show commands: SageMath
sage: E = EllipticCurve("dh1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 28224.dh

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
28224.dh1 28224bi6 \([0, 0, 0, -77066220, -260401928528]\) \(2251439055699625/25088\) \(564055361099661312\) \([2]\) \(1327104\) \(2.9751\)  
28224.dh2 28224bi5 \([0, 0, 0, -4812780, -4075624784]\) \(-548347731625/1835008\) \(-41256620697575227392\) \([2]\) \(663552\) \(2.6285\)  
28224.dh3 28224bi4 \([0, 0, 0, -1002540, -316696016]\) \(4956477625/941192\) \(21160889406254481408\) \([2]\) \(442368\) \(2.4258\)  
28224.dh4 28224bi2 \([0, 0, 0, -296940, 62239408]\) \(128787625/98\) \(2203341254295552\) \([2]\) \(147456\) \(1.8765\)  
28224.dh5 28224bi1 \([0, 0, 0, -14700, 1388464]\) \(-15625/28\) \(-629526072655872\) \([2]\) \(73728\) \(1.5299\) \(\Gamma_0(N)\)-optimal
28224.dh6 28224bi3 \([0, 0, 0, 126420, -29037008]\) \(9938375/21952\) \(-493548440962203648\) \([2]\) \(221184\) \(2.0792\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 28224.2.a.dh

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

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.