Properties

Label 28224.bd
Number of curves $4$
Conductor $28224$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("bd1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 28224.bd

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
28224.bd1 28224ck4 \([0, 0, 0, -527436, 147435120]\) \(1443468546/7\) \(78690759081984\) \([2]\) \(196608\) \(1.8668\)  
28224.bd2 28224ck3 \([0, 0, 0, -104076, -10224144]\) \(11090466/2401\) \(26990930365120512\) \([2]\) \(196608\) \(1.8668\)  
28224.bd3 28224ck2 \([0, 0, 0, -33516, 2222640]\) \(740772/49\) \(275417656786944\) \([2, 2]\) \(98304\) \(1.5202\)  
28224.bd4 28224ck1 \([0, 0, 0, 1764, 148176]\) \(432/7\) \(-9836344885248\) \([2]\) \(49152\) \(1.1737\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 28224.2.a.bd

sage: E.q_eigenform(10)
 
\(q - 2q^{5} - 4q^{11} + 2q^{13} - 6q^{17} + 8q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.