Properties

Label 99099.s
Number of curves $4$
Conductor $99099$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("s1")
 
E.isogeny_class()
 

Elliptic curves in class 99099.s

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
99099.s1 99099cd4 \([1, -1, 1, -10798184, -13654893392]\) \(107818231938348177/4463459\) \(5764414329444771\) \([2]\) \(2334720\) \(2.5105\)  
99099.s2 99099cd3 \([1, -1, 1, -1095194, 82959220]\) \(112489728522417/62811265517\) \(81118737507559724973\) \([2]\) \(2334720\) \(2.5105\)  
99099.s3 99099cd2 \([1, -1, 1, -675929, -212538752]\) \(26444947540257/169338169\) \(218694821192608761\) \([2, 2]\) \(1167360\) \(2.1639\)  
99099.s4 99099cd1 \([1, -1, 1, -17084, -7242650]\) \(-426957777/17320303\) \(-22368616537874607\) \([2]\) \(583680\) \(1.8173\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 99099.s have rank \(1\).

Complex multiplication

The elliptic curves in class 99099.s do not have complex multiplication.

Modular form 99099.2.a.s

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{4} + 2 q^{5} + q^{7} + 3 q^{8} - 2 q^{10} + q^{13} - q^{14} - q^{16} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.