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Properties

Label 235200.qn
Number of curves $6$
Conductor $235200$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("qn1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 235200.qn have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(5\)\(1\)
\(7\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 235200.qn do not have complex multiplication.

Modular form 235200.2.a.qn

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 235200.qn

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
235200.qn1 235200qn5 \([0, 1, 0, -119561633, 503031868863]\) \(784478485879202/221484375\) \(53365586400000000000000\) \([2]\) \(37748736\) \(3.3420\)  
235200.qn2 235200qn3 \([0, 1, 0, -8429633, 5716168863]\) \(549871953124/200930625\) \(24206629991040000000000\) \([2, 2]\) \(18874368\) \(2.9954\)  
235200.qn3 235200qn2 \([0, 1, 0, -3627633, -2596093137]\) \(175293437776/4862025\) \(146435169081600000000\) \([2, 2]\) \(9437184\) \(2.6488\)  
235200.qn4 235200qn1 \([0, 1, 0, -3603133, -2633700637]\) \(2748251600896/2205\) \(4150656720000000\) \([2]\) \(4718592\) \(2.3023\) \(\Gamma_0(N)\)-optimal
235200.qn5 235200qn4 \([0, 1, 0, 782367, -8501083137]\) \(439608956/259416045\) \(-31252519196881920000000\) \([2]\) \(18874368\) \(2.9954\)  
235200.qn6 235200qn6 \([0, 1, 0, 25870367, 40462068863]\) \(7947184069438/7533176175\) \(-1815082278528153600000000\) \([2]\) \(37748736\) \(3.3420\)