Properties

Label 25872bk
Number of curves $4$
Conductor $25872$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bk1")
 
E.isogeny_class()
 

Elliptic curves in class 25872bk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
25872.be3 25872bk1 \([0, -1, 0, -5112, -137808]\) \(30664297/297\) \(143121420288\) \([2]\) \(27648\) \(0.96032\) \(\Gamma_0(N)\)-optimal
25872.be2 25872bk2 \([0, -1, 0, -9032, 106800]\) \(169112377/88209\) \(42507061825536\) \([2, 2]\) \(55296\) \(1.3069\)  
25872.be4 25872bk3 \([0, -1, 0, 34088, 796720]\) \(9090072503/5845851\) \(-2817058915528704\) \([2]\) \(110592\) \(1.6535\)  
25872.be1 25872bk4 \([0, -1, 0, -114872, 15009072]\) \(347873904937/395307\) \(190494610403328\) \([2]\) \(110592\) \(1.6535\)  

Rank

sage: E.rank()
 

The elliptic curves in class 25872bk have rank \(1\).

Complex multiplication

The elliptic curves in class 25872bk do not have complex multiplication.

Modular form 25872.2.a.bk

sage: E.q_eigenform(10)
 
\(q - q^{3} + 2 q^{5} + q^{9} - q^{11} + 2 q^{13} - 2 q^{15} + 2 q^{17} + O(q^{20})\) Copy content 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.