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Properties

Label 327600.hl
Number of curves $6$
Conductor $327600$
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, 0, 0, -636854475, -6185978707750]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -636854475, -6185978707750]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -636854475, -6185978707750]) 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 327600.hl have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(5\)\(1\)
\(7\)\(1 - T\)
\(13\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(17\) \( 1 + 6 T + 17 T^{2}\) 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.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 327600.hl do not have complex multiplication.

Modular form 327600.2.a.hl

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);
 
\(q + q^{7} - 4 q^{11} - q^{13} - 6 q^{17} - 4 q^{19} + O(q^{20})\) 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 327600.hl

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
327600.hl1 327600hl6 \([0, 0, 0, -636854475, -6185978707750]\) \(1224522642327678150914/66339\) \(1547556192000000\) \([2]\) \(37748736\) \(3.3053\)  
327600.hl2 327600hl4 \([0, 0, 0, -39803475, -96655558750]\) \(597914615076708388/4400862921\) \(51331665110544000000\) \([2, 2]\) \(18874368\) \(2.9587\)  
327600.hl3 327600hl5 \([0, 0, 0, -38984475, -100823449750]\) \(-280880296871140514/25701087819771\) \(-599554976659617888000000\) \([2]\) \(37748736\) \(3.3053\)  
327600.hl4 327600hl3 \([0, 0, 0, -8492475, 7848652250]\) \(5807363790481348/1079211743883\) \(12587925780651312000000\) \([2]\) \(18874368\) \(2.9587\)  
327600.hl5 327600hl2 \([0, 0, 0, -2538975, -1444761250]\) \(620742479063632/49991146569\) \(145774183395204000000\) \([2, 2]\) \(9437184\) \(2.6122\)  
327600.hl6 327600hl1 \([0, 0, 0, 162150, -102302125]\) \(2587063175168/26304786963\) \(-4794047424006750000\) \([2]\) \(4718592\) \(2.2656\) \(\Gamma_0(N)\)-optimal