Properties

Label 10192bb
Number of curves $3$
Conductor $10192$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bb1")
 
E.isogeny_class()
 

Elliptic curves in class 10192bb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
10192.g2 10192bb1 \([0, 1, 0, -5749, 166179]\) \(-43614208/91\) \(-43852017664\) \([]\) \(13824\) \(0.92759\) \(\Gamma_0(N)\)-optimal
10192.g3 10192bb2 \([0, 1, 0, 9931, 837283]\) \(224755712/753571\) \(-363138558275584\) \([]\) \(41472\) \(1.4769\)  
10192.g1 10192bb3 \([0, 1, 0, -91989, -26497661]\) \(-178643795968/524596891\) \(-252798155281444864\) \([]\) \(124416\) \(2.0262\)  

Rank

sage: E.rank()
 

The elliptic curves in class 10192bb have rank \(1\).

Complex multiplication

The elliptic curves in class 10192bb do not have complex multiplication.

Modular form 10192.2.a.bb

sage: E.q_eigenform(10)
 
\(q - 2 q^{3} + 3 q^{5} + q^{9} - q^{13} - 6 q^{15} + 6 q^{17} - 7 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.