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Properties

Label 193200he
Number of curves $6$
Conductor $193200$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("he1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 193200he have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(5\)\(1\)
\(7\)\(1 - T\)
\(23\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 - 3 T + 11 T^{2}\) 1.11.ad
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(29\) \( 1 + 29 T^{2}\) 1.29.a
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 193200.2.a.he

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} - q^{7} + q^{9} - 4 q^{11} - 6 q^{13} - 2 q^{17} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 193200he

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
193200.d4 193200he1 \([0, -1, 0, -2314383, -1354421238]\) \(5483900709072173056/277725\) \(69431250000\) \([2]\) \(1966080\) \(2.0018\) \(\Gamma_0(N)\)-optimal
193200.d3 193200he2 \([0, -1, 0, -2314508, -1354267488]\) \(342799332162880336/77131175625\) \(308524702500000000\) \([2, 2]\) \(3932160\) \(2.3484\)  
193200.d2 193200he3 \([0, -1, 0, -2579008, -1025229488]\) \(118566490663726564/40187675390625\) \(643002806250000000000\) \([2, 2]\) \(7864320\) \(2.6949\)  
193200.d5 193200he4 \([0, -1, 0, -2052008, -1673467488]\) \(-59722927783102084/41113267272525\) \(-657812276360400000000\) \([2]\) \(7864320\) \(2.6949\)  
193200.d1 193200he5 \([0, -1, 0, -16954008, 26114770512]\) \(16841893263968213282/543703603314375\) \(17398515306060000000000\) \([2]\) \(15728640\) \(3.0415\)  
193200.d6 193200he6 \([0, -1, 0, 7563992, -7111029488]\) \(1495639267637215678/1547698974609375\) \(-49526367187500000000000\) \([2]\) \(15728640\) \(3.0415\)