Properties

Label 193200ft
Number of curves $6$
Conductor $193200$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("ft1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 193200ft

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
193200.hl6 193200ft1 \([0, 1, 0, 57617, -1636012]\) \(84611246065664/53699121315\) \(-13424780328750000\) \([2]\) \(1376256\) \(1.7837\) \(\Gamma_0(N)\)-optimal
193200.hl5 193200ft2 \([0, 1, 0, -242508, -13641012]\) \(394315384276816/208332909225\) \(833331636900000000\) \([2, 2]\) \(2752512\) \(2.1302\)  
193200.hl3 193200ft3 \([0, 1, 0, -2227008, 1268345988]\) \(76343005935514084/694180580625\) \(11106889290000000000\) \([2, 2]\) \(5505024\) \(2.4768\)  
193200.hl2 193200ft4 \([0, 1, 0, -3060008, -2059146012]\) \(198048499826486404/242568272835\) \(3881092365360000000\) \([2]\) \(5505024\) \(2.4768\)  
193200.hl1 193200ft5 \([0, 1, 0, -35554008, 81586415988]\) \(155324313723954725282/13018359375\) \(416587500000000000\) \([2]\) \(11010048\) \(2.8234\)  
193200.hl4 193200ft6 \([0, 1, 0, -652008, 3029195988]\) \(-957928673903042/123339801817575\) \(-3946873658162400000000\) \([2]\) \(11010048\) \(2.8234\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 193200.2.a.ft

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