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Properties

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

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Show commands: SageMath
Copy content sage:E = EllipticCurve("p1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 106470.p 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
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 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.p do not have complex multiplication.

Modular form 106470.2.a.p

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 106470.p

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
106470.p1 106470w7 \([1, -1, 0, -3446097030, 77852403587700]\) \(1286229821345376481036009/247265484375000000\) \(870063880455174234375000000\) \([2]\) \(111476736\) \(4.1697\)  
106470.p2 106470w8 \([1, -1, 0, -1515765510, -22000772931084]\) \(109454124781830273937129/3914078300576808000\) \(13772638600220125851294888000\) \([2]\) \(111476736\) \(4.1697\)  
106470.p3 106470w5 \([1, -1, 0, -1502509995, -22416452817615]\) \(106607603143751752938169/5290068420\) \(18614395248138127620\) \([2]\) \(37158912\) \(3.6204\)  
106470.p4 106470w6 \([1, -1, 0, -238125510, 943852772916]\) \(424378956393532177129/136231857216000000\) \(479364997628242829376000000\) \([2, 2]\) \(55738368\) \(3.8231\)  
106470.p5 106470w4 \([1, -1, 0, -104589315, -265615756119]\) \(35958207000163259449/12145729518877500\) \(42737709967343734848277500\) \([2]\) \(37158912\) \(3.6204\)  
106470.p6 106470w2 \([1, -1, 0, -93911895, -350200141875]\) \(26031421522845051769/5797789779600\) \(20400936614557065075600\) \([2, 2]\) \(18579456\) \(3.2738\)  
106470.p7 106470w1 \([1, -1, 0, -5207175, -6753206979]\) \(-4437543642183289/3033210136320\) \(-10673089242977959499520\) \([2]\) \(9289728\) \(2.9273\) \(\Gamma_0(N)\)-optimal
106470.p8 106470w3 \([1, -1, 0, 42225210, 100613877300]\) \(2366200373628880151/2612420149248000\) \(-9192437101277088841728000\) \([2]\) \(27869184\) \(3.4766\)