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Properties

Label 182070dj
Number of curves $8$
Conductor $182070$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("dj1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 182070dj have rank \(0\).

L-function data

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

Complex multiplication

The elliptic curves in class 182070dj do not have complex multiplication.

Modular form 182070.2.a.dj

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{5} - q^{7} - q^{8} + q^{10} + 2 q^{13} + q^{14} + q^{16} - 4 q^{19} + O(q^{20})\) 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 182070dj

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
182070.l7 182070dj1 \([1, -1, 0, -106695, 4746685]\) \(7633736209/3870720\) \(68110303117086720\) \([2]\) \(1769472\) \(1.9225\) \(\Gamma_0(N)\)-optimal
182070.l5 182070dj2 \([1, -1, 0, -939015, -346658819]\) \(5203798902289/57153600\) \(1005691194463233600\) \([2, 2]\) \(3538944\) \(2.2691\)  
182070.l4 182070dj3 \([1, -1, 0, -6973335, 7089496141]\) \(2131200347946769/2058000\) \(36213160294458000\) \([2]\) \(5308416\) \(2.4718\)  
182070.l6 182070dj4 \([1, -1, 0, -210735, -871166075]\) \(-58818484369/18600435000\) \(-327298607483793435000\) \([2]\) \(7077888\) \(2.6156\)  
182070.l2 182070dj5 \([1, -1, 0, -14984415, -22322091659]\) \(21145699168383889/2593080\) \(45628581971017080\) \([2]\) \(7077888\) \(2.6156\)  
182070.l3 182070dj6 \([1, -1, 0, -7025355, 6978391825]\) \(2179252305146449/66177562500\) \(1164479435718665062500\) \([2, 2]\) \(10616832\) \(2.8184\)  
182070.l8 182070dj7 \([1, -1, 0, 1896075, 23484821611]\) \(42841933504271/13565917968750\) \(-238709796862882324218750\) \([2]\) \(21233664\) \(3.1649\)  
182070.l1 182070dj8 \([1, -1, 0, -16779105, -16639338425]\) \(29689921233686449/10380965400750\) \(182666454843820301250750\) \([2]\) \(21233664\) \(3.1649\)