Properties

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

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Show commands for: SageMath

sage: E = EllipticCurve("28224.fk1")
sage: E.isogeny_class()

Elliptic curves in class 28224bz

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
28224.fk6 28224bz1 [0, 0, 0, 27636, -389648] 2 98304 \(\Gamma_0(N)\)-optimal
28224.fk5 28224bz2 [0, 0, 0, -113484, -3155600] 4 196608  
28224.fk3 28224bz3 [0, 0, 0, -1101324, 442162672] 2 393216  
28224.fk2 28224bz4 [0, 0, 0, -1383564, -625494800] 4 393216  
28224.fk4 28224bz5 [0, 0, 0, -960204, -1015494032] 2 786432  
28224.fk1 28224bz6 [0, 0, 0, -22128204, -40065204368] 2 786432  

Rank

sage: E.rank()

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

Modular form 28224.2.a.fk

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

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 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

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

The vertices are labelled with Cremona labels.