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Properties

Label 163254i
Number of curves $4$
Conductor $163254$
CM no
Rank $2$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 0, 784917, 1885107249]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, 784917, 1885107249]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, 784917, 1885107249]) 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 163254i have rank \(2\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 + T\)
\(7\)\(1 - T\)
\(13\)\(1\)
\(23\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + T + 5 T^{2}\) 1.5.b
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 + 5 T + 19 T^{2}\) 1.19.f
\(29\) \( 1 - 8 T + 29 T^{2}\) 1.29.ai
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 163254i do not have complex multiplication.

Modular form 163254.2.a.i

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);
 
\(q + q^{2} + q^{3} + q^{4} + q^{6} - q^{7} + q^{8} + q^{9} - 6 q^{11} + q^{12} - q^{14} + q^{16} - 6 q^{17} + q^{18} - 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

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 163254i

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
163254.dj4 163254i1 \([1, 0, 0, 784917, 1885107249]\) \(11079872671250375/324440155855872\) \(-1566010664246525672448\) \([2]\) \(11059200\) \(2.7470\) \(\Gamma_0(N)\)-optimal
163254.dj2 163254i2 \([1, 0, 0, -18927243, 30195711441]\) \(155355156733986861625/8291568305839392\) \(40021816522740329860128\) \([2]\) \(22118400\) \(3.0935\)  
163254.dj3 163254i3 \([1, 0, 0, -7086258, -51790953852]\) \(-8152944444844179625/235342826399858688\) \(-1135954872552275513966592\) \([2]\) \(33177600\) \(3.2963\)  
163254.dj1 163254i4 \([1, 0, 0, -256286898, -1571565976956]\) \(385693937170561837203625/2159357734550274048\) \(10422807347346873727352832\) \([2]\) \(66355200\) \(3.6428\)