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Properties

Label 106470fe
Number of curves $8$
Conductor $106470$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("fe1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 106470fe have rank \(0\).

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 \(\mathbb{F}_p\)
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(17\) \( 1 + 4 T + 17 T^{2}\) 1.17.e
\(19\) \( 1 - 2 T + 19 T^{2}\) 1.19.ac
\(23\) \( 1 - 2 T + 23 T^{2}\) 1.23.ac
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 106470fe do not have complex multiplication.

Modular form 106470.2.a.fe

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + q^{5} - q^{7} + q^{8} + q^{10} - q^{14} + q^{16} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 106470fe

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
106470.fd7 106470fe1 \([1, -1, 1, -39127757, -93690555019]\) \(1882742462388824401/11650189824000\) \(40994032757665688064000\) \([2]\) \(12386304\) \(3.1771\) \(\Gamma_0(N)\)-optimal
106470.fd6 106470fe2 \([1, -1, 1, -62977037, 34036648949]\) \(7850236389974007121/4400862921000000\) \(15485508946284985881000000\) \([2, 2]\) \(24772608\) \(3.5237\)  
106470.fd5 106470fe3 \([1, -1, 1, -241724957, 1383584710421]\) \(443915739051786565201/21894701746029840\) \(77041845167798306019828240\) \([2]\) \(37158912\) \(3.7264\)  
106470.fd8 106470fe4 \([1, -1, 1, 247489483, 269867017541]\) \(476437916651992691759/284661685546875000\) \(-1001651530013810279296875000\) \([2]\) \(49545216\) \(3.8703\)  
106470.fd4 106470fe5 \([1, -1, 1, -755032037, 7972461142949]\) \(13527956825588849127121/25701087819771000\) \(90435542416732298698731000\) \([2]\) \(49545216\) \(3.8703\)  
106470.fd2 106470fe6 \([1, -1, 1, -3820607537, 90897164248349]\) \(1752803993935029634719121/4599740941532100\) \(16185309740232342656228100\) \([2, 2]\) \(74317824\) \(4.0730\)  
106470.fd3 106470fe7 \([1, -1, 1, -3773654267, 93240113640041]\) \(-1688971789881664420008241/89901485966373558750\) \(-316340292848806015646929458750\) \([2]\) \(148635648\) \(4.4196\)  
106470.fd1 106470fe8 \([1, -1, 1, -61129682087, 5817380358693089]\) \(7179471593960193209684686321/49441793310\) \(173973001742214038910\) \([2]\) \(148635648\) \(4.4196\)