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Properties

Label 338130y
Number of curves $8$
Conductor $338130$
CM no
Rank $1$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, -1, 0, -5452075800, -149202592145600]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 0, -5452075800, -149202592145600]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 0, -5452075800, -149202592145600]) ellisomat(E)
 

Rank

Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 

The elliptic curves in class 338130y have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1\)
\(5\)\(1 + T\)
\(13\)\(1 - T\)
\(17\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 - 4 T + 7 T^{2}\) 1.7.ae
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 338130y do not have complex multiplication.

Modular form 338130.2.a.y

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 
\(q - q^{2} + q^{4} - q^{5} + 4 q^{7} - q^{8} + q^{10} + q^{13} - 4 q^{14} + q^{16} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 
Copy content gp:ellisomat(E)
 

The \((i,j)\)-th 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 comment:Isogeny graph
 
Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 338130y

Copy content comment:List of curves in the isogeny class
 
Copy content sage:E.isogeny_class().curves
 
Copy content magma:IsogenousCurves(E);
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
338130.y7 338130y1 \([1, -1, 0, -5452075800, -149202592145600]\) \(1018563973439611524445729/42904970360310988800\) \(754968206553406826787687628800\) \([2]\) \(637009920\) \(4.4977\) \(\Gamma_0(N)\)-optimal
338130.y6 338130y2 \([1, -1, 0, -14454448920, 470482963468096]\) \(18980483520595353274840609/5549773448629762560000\) \(97655410832437591100054530560000\) \([2, 2]\) \(1274019840\) \(4.8442\)  
338130.y5 338130y3 \([1, -1, 0, -67087036440, 6644606911560256]\) \(1897660325010178513043539489/14258428094958372000000\) \(250895404348751670826419972000000\) \([2]\) \(1911029760\) \(5.0470\)  
338130.y4 338130y4 \([1, -1, 0, -211651864920, 37473670967754496]\) \(59589391972023341137821784609/8834417507562311995200\) \(155452953017259535808493730555200\) \([2]\) \(2548039680\) \(5.1908\)  
338130.y8 338130y5 \([1, -1, 0, 38704997160, 3127062489980800]\) \(364421318680576777174674911/450962301637624725000000\) \(-7935262447017018271133479725000000\) \([2]\) \(2548039680\) \(5.1908\)  
338130.y2 338130y6 \([1, -1, 0, -1071449453160, 426880086563858800]\) \(7730680381889320597382223137569/441370202660156250000\) \(7766477112793805170628906250000\) \([2, 2]\) \(3822059520\) \(5.3935\)  
338130.y1 338130y7 \([1, -1, 0, -17143191015660, 27320313468836296300]\) \(31664865542564944883878115208137569/103216295812500\) \(1816223646869901133312500\) \([2]\) \(7644119040\) \(5.7401\)  
338130.y3 338130y8 \([1, -1, 0, -1069506558180, 428505349689527476]\) \(-7688701694683937879808871873249/58423707246780395507812500\) \(-1028040367115717170000076293945312500\) \([2]\) \(7644119040\) \(5.7401\)