Properties

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

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Show commands for: SageMath
sage: E = EllipticCurve("d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 192d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
192.b5 192d1 \([0, -1, 0, 3, -3]\) \(2048/3\) \(-3072\) \([2]\) \(8\) \(-0.64535\) \(\Gamma_0(N)\)-optimal
192.b4 192d2 \([0, -1, 0, -17, -15]\) \(35152/9\) \(147456\) \([2, 2]\) \(16\) \(-0.29878\)  
192.b2 192d3 \([0, -1, 0, -257, -1503]\) \(28756228/3\) \(196608\) \([2]\) \(32\) \(0.047795\)  
192.b3 192d4 \([0, -1, 0, -97, 385]\) \(1556068/81\) \(5308416\) \([2, 2]\) \(32\) \(0.047795\)  
192.b1 192d5 \([0, -1, 0, -1537, 23713]\) \(3065617154/9\) \(1179648\) \([4]\) \(64\) \(0.39437\)  
192.b6 192d6 \([0, -1, 0, 63, 1377]\) \(207646/6561\) \(-859963392\) \([2]\) \(64\) \(0.39437\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 192.2.a.d

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.