Properties

Label 193200bk
Number of curves $4$
Conductor $193200$
CM no
Rank $1$
Graph

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

Elliptic curves in class 193200bk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
193200.gv3 193200bk1 \([0, 1, 0, -50849408, -149494228812]\) \(-227196402372228188089/19338934824115200\) \(-1237691828743372800000000\) \([2]\) \(26542080\) \(3.3678\) \(\Gamma_0(N)\)-optimal
193200.gv2 193200bk2 \([0, 1, 0, -829537408, -9196291412812]\) \(986396822567235411402169/6336721794060000\) \(405550194819840000000000\) \([2]\) \(53084160\) \(3.7143\)  
193200.gv4 193200bk3 \([0, 1, 0, 301464592, -5426368812]\) \(47342661265381757089751/27397579603968000000\) \(-1753445094653952000000000000\) \([2]\) \(79626240\) \(3.9171\)  
193200.gv1 193200bk4 \([0, 1, 0, -1205863408, -44616896812]\) \(3029968325354577848895529/1753440696000000000000\) \(112220204544000000000000000000\) \([2]\) \(159252480\) \(4.2637\)  

Rank

sage: E.rank()
 

The elliptic curves in class 193200bk have rank \(1\).

Complex multiplication

The elliptic curves in class 193200bk do not have complex multiplication.

Modular form 193200.2.a.bk

sage: E.q_eigenform(10)
 
\(q + q^{3} + q^{7} + q^{9} + 4q^{13} - 2q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.