Properties

Label 193600ge
Number of curves $4$
Conductor $193600$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("ge1")
 
E.isogeny_class()
 

Elliptic curves in class 193600ge

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
193600.u4 193600ge1 \([0, 1, 0, -548533, -153346437]\) \(643956736/15125\) \(428717762000000000\) \([2]\) \(3317760\) \(2.1688\) \(\Gamma_0(N)\)-optimal
193600.u3 193600ge2 \([0, 1, 0, -1214033, 290542063]\) \(436334416/171875\) \(77948684000000000000\) \([2]\) \(6635520\) \(2.5154\)  
193600.u2 193600ge3 \([0, 1, 0, -5388533, 4751993563]\) \(610462990336/8857805\) \(251074270137680000000\) \([2]\) \(9953280\) \(2.7181\)  
193600.u1 193600ge4 \([0, 1, 0, -85914033, 306481042063]\) \(154639330142416/33275\) \(15090865222400000000\) \([2]\) \(19906560\) \(3.0647\)  

Rank

sage: E.rank()
 

The elliptic curves in class 193600ge have rank \(2\).

Complex multiplication

The elliptic curves in class 193600ge do not have complex multiplication.

Modular form 193600.2.a.ge

sage: E.q_eigenform(10)
 
\(q - 2 q^{3} - 4 q^{7} + q^{9} + 4 q^{13} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.