Properties

Label 192c
Number of curves $6$
Conductor $192$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("c1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 192c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
192.d5 192c1 \([0, 1, 0, 3, 3]\) \(2048/3\) \(-3072\) \([2]\) \(8\) \(-0.64535\) \(\Gamma_0(N)\)-optimal
192.d4 192c2 \([0, 1, 0, -17, 15]\) \(35152/9\) \(147456\) \([2, 2]\) \(16\) \(-0.29878\)  
192.d3 192c3 \([0, 1, 0, -97, -385]\) \(1556068/81\) \(5308416\) \([2, 2]\) \(32\) \(0.047795\)  
192.d2 192c4 \([0, 1, 0, -257, 1503]\) \(28756228/3\) \(196608\) \([2]\) \(32\) \(0.047795\)  
192.d1 192c5 \([0, 1, 0, -1537, -23713]\) \(3065617154/9\) \(1179648\) \([2]\) \(64\) \(0.39437\)  
192.d6 192c6 \([0, 1, 0, 63, -1377]\) \(207646/6561\) \(-859963392\) \([4]\) \(64\) \(0.39437\)  

Rank

sage: E.rank()
 

The elliptic curves in class 192c have rank \(0\).

Complex multiplication

The elliptic curves in class 192c do not have complex multiplication.

Modular form 192.2.a.c

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

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.