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Properties

Label 4368.p
Number of curves $6$
Conductor $4368$
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([0, 1, 0, -2830464, 1831939092]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 0, -2830464, 1831939092]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 0, -2830464, 1831939092]) 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 4368.p have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(7\)\(1 + T\)
\(13\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 + 4 T + 19 T^{2}\) 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 4368.p do not have complex multiplication.

Modular form 4368.2.a.p

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);
 
\(q + q^{3} - 2 q^{5} - q^{7} + q^{9} + 4 q^{11} + q^{13} - 2 q^{15} - 6 q^{17} - 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

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

Elliptic curves in class 4368.p

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
4368.p1 4368k5 \([0, 1, 0, -2830464, 1831939092]\) \(1224522642327678150914/66339\) \(135862272\) \([4]\) \(36864\) \(1.9513\)  
4368.p2 4368k4 \([0, 1, 0, -176904, 28579716]\) \(597914615076708388/4400862921\) \(4506483631104\) \([2, 4]\) \(18432\) \(1.6047\)  
4368.p3 4368k6 \([0, 1, 0, -173264, 29815860]\) \(-280880296871140514/25701087819771\) \(-52635827854891008\) \([4]\) \(36864\) \(1.9513\)  
4368.p4 4368k3 \([0, 1, 0, -37744, -2338108]\) \(5807363790481348/1079211743883\) \(1105112825736192\) \([2]\) \(18432\) \(1.6047\)  
4368.p5 4368k2 \([0, 1, 0, -11284, 424316]\) \(620742479063632/49991146569\) \(12797733521664\) \([2, 2]\) \(9216\) \(1.2581\)  
4368.p6 4368k1 \([0, 1, 0, 721, 30552]\) \(2587063175168/26304786963\) \(-420876591408\) \([2]\) \(4608\) \(0.91157\) \(\Gamma_0(N)\)-optimal