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Properties

Label 19800.w
Number of curves $6$
Conductor $19800$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("w1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 19800.w have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(5\)\(1\)
\(11\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 19800.w do not have complex multiplication.

Modular form 19800.2.a.w

Copy content sage:E.q_eigenform(10)
 
\(q + q^{11} + 2 q^{13} - 6 q^{17} - 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 19800.w

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
19800.w1 19800bi3 \([0, 0, 0, -11880075, -15760750250]\) \(15897679904620804/2475\) \(28868400000000\) \([2]\) \(393216\) \(2.4294\)  
19800.w2 19800bi5 \([0, 0, 0, -6300075, 5969839750]\) \(1185450336504002/26043266205\) \(607537314030240000000\) \([2]\) \(786432\) \(2.7760\)  
19800.w3 19800bi4 \([0, 0, 0, -855075, -166675250]\) \(5927735656804/2401490025\) \(28010979651600000000\) \([2, 2]\) \(393216\) \(2.4294\)  
19800.w4 19800bi2 \([0, 0, 0, -742575, -246212750]\) \(15529488955216/6125625\) \(17862322500000000\) \([2, 2]\) \(196608\) \(2.0828\)  
19800.w5 19800bi1 \([0, 0, 0, -39450, -5040875]\) \(-37256083456/38671875\) \(-7047949218750000\) \([2]\) \(98304\) \(1.7363\) \(\Gamma_0(N)\)-optimal
19800.w6 19800bi6 \([0, 0, 0, 2789925, -1212790250]\) \(102949393183198/86815346805\) \(-2025228410267040000000\) \([2]\) \(786432\) \(2.7760\)