Properties

Label 247962bk
Number of curves $4$
Conductor $247962$
CM no
Rank $0$
Graph

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

Elliptic curves in class 247962bk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
247962.bk4 247962bk1 \([1, 0, 0, 258752098, -175908011132220]\) \(79374649975090937760383/553856914190911653543936\) \(-13368759482410209210320849731584\) \([2]\) \(680804352\) \(4.6515\) \(\Gamma_0(N)\)-optimal
247962.bk3 247962bk2 \([1, 0, 0, -45575815582, -3672581344871740]\) \(433744050935826360922067531137/9612122270219882316693504\) \(232013264533869054591069304651776\) \([2, 2]\) \(1361608704\) \(4.9980\)  
247962.bk2 247962bk3 \([1, 0, 0, -99214955422, 6615094569429188]\) \(4474676144192042711273397261697/1806328356954994499451382272\) \(43600375352657809625128201735776768\) \([2]\) \(2723217408\) \(5.3446\)  
247962.bk1 247962bk4 \([1, 0, 0, -725289758622, -237747167280231228]\) \(1748094148784980747354970849498497/887694600425282263291392\) \(21426789668692679974672141506048\) \([2]\) \(2723217408\) \(5.3446\)  

Rank

sage: E.rank()
 

The elliptic curves in class 247962bk have rank \(0\).

Complex multiplication

The elliptic curves in class 247962bk do not have complex multiplication.

Modular form 247962.2.a.bk

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} + 2q^{5} + q^{6} + q^{8} + q^{9} + 2q^{10} - q^{11} + q^{12} + q^{13} + 2q^{15} + q^{16} + q^{18} + 8q^{19} + O(q^{20})\)  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 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.