Properties

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

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Show commands: SageMath
E = EllipticCurve("cg1")
 
E.isogeny_class()
 

Elliptic curves in class 193200cg

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
193200.eq4 193200cg1 \([0, 1, 0, -456408, -355480812]\) \(-164287467238609/757170892800\) \(-48458937139200000000\) \([2]\) \(5308416\) \(2.4620\) \(\Gamma_0(N)\)-optimal
193200.eq3 193200cg2 \([0, 1, 0, -10824408, -13688728812]\) \(2191574502231419089/4115217960000\) \(263373949440000000000\) \([2, 2]\) \(10616832\) \(2.8086\)  
193200.eq2 193200cg3 \([0, 1, 0, -14424408, -3803128812]\) \(5186062692284555089/2903809817953800\) \(185843828349043200000000\) \([2]\) \(21233664\) \(3.1551\)  
193200.eq1 193200cg4 \([0, 1, 0, -173112408, -876736312812]\) \(8964546681033941529169/31696875000\) \(2028600000000000000\) \([2]\) \(21233664\) \(3.1551\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 193200.2.a.cg

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.