Properties

Label 229320.er
Number of curves $6$
Conductor $229320$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
sage: E = EllipticCurve("er1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 229320.er

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
229320.er1 229320dj3 \([0, 0, 0, -89893587, 328050858814]\) \(914732517663095044/9555\) \(839163173022720\) \([2]\) \(12582912\) \(2.8915\)  
229320.er2 229320dj6 \([0, 0, 0, -33833667, -72217661474]\) \(24385137179326562/1284775885575\) \(225669619835100374169600\) \([2]\) \(25165824\) \(3.2381\)  
229320.er3 229320dj4 \([0, 0, 0, -6050667, 4291163926]\) \(278944461825124/70849130625\) \(6222290032566840960000\) \([2, 2]\) \(12582912\) \(2.8915\)  
229320.er4 229320dj2 \([0, 0, 0, -5618487, 5125530634]\) \(893359210685776/91298025\) \(2004551029558022400\) \([2, 2]\) \(6291456\) \(2.5450\)  
229320.er5 229320dj1 \([0, 0, 0, -324282, 92859361]\) \(-2748251600896/1124136195\) \(-1542604814733593520\) \([2]\) \(3145728\) \(2.1984\) \(\Gamma_0(N)\)-optimal
229320.er6 229320dj5 \([0, 0, 0, 14817453, 27400520014]\) \(2048324060764798/3031899609375\) \(-532550182415378400000000\) \([2]\) \(25165824\) \(3.2381\)  

Rank

sage: E.rank()
 

The elliptic curves in class 229320.er have rank \(0\).

Complex multiplication

The elliptic curves in class 229320.er do not have complex multiplication.

Modular form 229320.2.a.er

sage: E.q_eigenform(10)
 
\(q + q^{5} + 4q^{11} - q^{13} + 2q^{17} - 4q^{19} + O(q^{20})\)  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 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

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

The vertices are labelled with LMFDB labels.