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Properties

Label 106470ea
Number of curves $8$
Conductor $106470$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ea1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 106470ea 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 \(\mathbb{F}_p\)
\(11\) \( 1 + 6 T + 11 T^{2}\) 1.11.g
\(17\) \( 1 - 8 T + 17 T^{2}\) 1.17.ai
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 + 23 T^{2}\) 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 106470ea do not have complex multiplication.

Modular form 106470.2.a.ea

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 106470ea

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
106470.dr7 106470ea1 \([1, -1, 1, -62393, 2126297]\) \(7633736209/3870720\) \(13620071850577920\) \([2]\) \(884736\) \(1.7884\) \(\Gamma_0(N)\)-optimal
106470.dr5 106470ea2 \([1, -1, 1, -549113, -154986919]\) \(5203798902289/57153600\) \(201108873418689600\) \([2, 2]\) \(1769472\) \(2.1349\)  
106470.dr4 106470ea3 \([1, -1, 1, -4077833, 3170527481]\) \(2131200347946769/2058000\) \(7241574660138000\) \([2]\) \(2654208\) \(2.3377\)  
106470.dr6 106470ea4 \([1, -1, 1, -123233, -389561623]\) \(-58818484369/18600435000\) \(-65450164608136035000\) \([2]\) \(3538944\) \(2.4815\)  
106470.dr2 106470ea5 \([1, -1, 1, -8762513, -9981498679]\) \(21145699168383889/2593080\) \(9124384071773880\) \([2]\) \(3538944\) \(2.4815\)  
106470.dr3 106470ea6 \([1, -1, 1, -4108253, 3120845537]\) \(2179252305146449/66177562500\) \(232861885165062562500\) \([2, 2]\) \(5308416\) \(2.6842\)  
106470.dr8 106470ea7 \([1, -1, 1, 1108777, 10501899581]\) \(42841933504271/13565917968750\) \(-47734989214776855468750\) \([2]\) \(10616832\) \(3.0308\)  
106470.dr1 106470ea8 \([1, -1, 1, -9812003, -7440217963]\) \(29689921233686449/10380965400750\) \(36527957237045927220750\) \([2]\) \(10616832\) \(3.0308\)