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Properties

Label 254898de
Number of curves $6$
Conductor $254898$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("de1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 254898de have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1\)
\(7\)\(1\)
\(17\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 254898de do not have complex multiplication.

Modular form 254898.2.a.de

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + 2 q^{5} - q^{8} - 2 q^{10} + 4 q^{11} + 2 q^{13} + q^{16} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 254898de

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
254898.de5 254898de1 \([1, -1, 0, -8952530211, 325631283424789]\) \(38331145780597164097/55468445663232\) \(114829980988643259739136851968\) \([2]\) \(424673280\) \(4.4783\) \(\Gamma_0(N)\)-optimal
254898.de4 254898de2 \([1, -1, 0, -11562685731, 120273121639957]\) \(82582985847542515777/44772582831427584\) \(92687559095480860975234438434816\) \([2, 2]\) \(849346560\) \(4.8249\)  
254898.de6 254898de3 \([1, -1, 0, 44596441629, 945935843912149]\) \(4738217997934888496063/2928751705237796928\) \(-6063059792134022817162450820663872\) \([2]\) \(1698693120\) \(5.1714\)  
254898.de2 254898de4 \([1, -1, 0, -109484301411, -13848343276727723]\) \(70108386184777836280897/552468975892674624\) \(1143713353421458698459693230131776\) \([2, 2]\) \(1698693120\) \(5.1714\)  
254898.de3 254898de5 \([1, -1, 0, -37292089851, -31838079298229555]\) \(-2770540998624539614657/209924951154647363208\) \(-434583624291257044043654616756155592\) \([2]\) \(3397386240\) \(5.5180\)  
254898.de1 254898de6 \([1, -1, 0, -1748422363851, -889851231810145571]\) \(285531136548675601769470657/17941034271597192\) \(37141271937419963029081880824008\) \([2]\) \(3397386240\) \(5.5180\)