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Properties

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

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bz1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

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

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(7\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 - 6 T + 13 T^{2}\) 1.13.ag
\(17\) \( 1 + 4 T + 17 T^{2}\) 1.17.e
\(19\) \( 1 - 6 T + 19 T^{2}\) 1.19.ag
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 28224.2.a.bz

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

Isogeny matrix

Copy content 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 28224bz

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
28224.fk6 28224bz1 \([0, 0, 0, 27636, -389648]\) \(103823/63\) \(-1416433663475712\) \([2]\) \(98304\) \(1.5965\) \(\Gamma_0(N)\)-optimal
28224.fk5 28224bz2 \([0, 0, 0, -113484, -3155600]\) \(7189057/3969\) \(89235320798969856\) \([2, 2]\) \(196608\) \(1.9430\)  
28224.fk3 28224bz3 \([0, 0, 0, -1101324, 442162672]\) \(6570725617/45927\) \(1032580140673794048\) \([2]\) \(393216\) \(2.2896\)  
28224.fk2 28224bz4 \([0, 0, 0, -1383564, -625494800]\) \(13027640977/21609\) \(485836746572169216\) \([2, 2]\) \(393216\) \(2.2896\)  
28224.fk4 28224bz5 \([0, 0, 0, -960204, -1015494032]\) \(-4354703137/17294403\) \(-388831342839926095872\) \([2]\) \(786432\) \(2.6362\)  
28224.fk1 28224bz6 \([0, 0, 0, -22128204, -40065204368]\) \(53297461115137/147\) \(3305011881443328\) \([2]\) \(786432\) \(2.6362\)