Properties

Label 3192.b
Number of curves $4$
Conductor $3192$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 3192.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3192.b1 3192c3 \([0, -1, 0, -592824, 175883580]\) \(22501000029889239268/3620708343\) \(3707605343232\) \([4]\) \(24576\) \(1.8142\)  
3192.b2 3192c2 \([0, -1, 0, -37164, 2739924]\) \(22174957026242512/278654127129\) \(71335456545024\) \([2, 2]\) \(12288\) \(1.4676\)  
3192.b3 3192c4 \([0, -1, 0, -6384, 7110684]\) \(-28104147578308/21301741002339\) \(-21812982786395136\) \([2]\) \(24576\) \(1.8142\)  
3192.b4 3192c1 \([0, -1, 0, -4359, -41940]\) \(572616640141312/280535480757\) \(4488567692112\) \([2]\) \(6144\) \(1.1210\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 3192.b have rank \(0\).

Complex multiplication

The elliptic curves in class 3192.b do not have complex multiplication.

Modular form 3192.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{5} - q^{7} + q^{9} - 4 q^{11} + 2 q^{13} + 2 q^{15} - 6 q^{17} + q^{19} + O(q^{20})\) 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 LMFDB 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 LMFDB labels.