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Properties

Label 106470.fo
Number of curves $6$
Conductor $106470$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("fo1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 106470.fo have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1\)
\(5\)\(1 - T\)
\(7\)\(1 - T\)
\(13\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 4 T + 19 T^{2}\) 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 106470.fo do not have complex multiplication.

Modular form 106470.2.a.fo

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + q^{5} + q^{7} + q^{8} + q^{10} - 4 q^{11} + q^{14} + q^{16} - 2 q^{17} - 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 106470.fo

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
106470.fo1 106470gb6 \([1, -1, 1, -93031997, 345402761019]\) \(25306558948218234961/4478906250\) \(15760123423291406250\) \([2]\) \(11010048\) \(3.0802\)  
106470.fo2 106470gb4 \([1, -1, 1, -5833067, 5361813591]\) \(6237734630203441/82168222500\) \(289128920274334822500\) \([2, 2]\) \(5505024\) \(2.7336\)  
106470.fo3 106470gb5 \([1, -1, 1, -889817, 14154866691]\) \(-22143063655441/24584858584650\) \(-86507817759804477868650\) \([2]\) \(11010048\) \(3.0802\)  
106470.fo4 106470gb2 \([1, -1, 1, -692087, -89681601]\) \(10418796526321/5038160400\) \(17727995474417264400\) \([2, 2]\) \(2752512\) \(2.3871\)  
106470.fo5 106470gb1 \([1, -1, 1, -570407, -165561249]\) \(5832972054001/4542720\) \(15984667657969920\) \([2]\) \(1376256\) \(2.0405\) \(\Gamma_0(N)\)-optimal
106470.fo6 106470gb3 \([1, -1, 1, 2502013, -686339481]\) \(492271755328079/342606902820\) \(-1205545901773408306020\) \([2]\) \(5505024\) \(2.7336\)