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Properties

Label 327600t
Number of curves $2$
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, -24003615, 45264946750]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -24003615, 45264946750]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -24003615, 45264946750]) 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 327600t 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
\(19\) \( 1 + 2 T + 19 T^{2}\) 1.19.c
\(23\) \( 1 + 2 T + 23 T^{2}\) 1.23.c
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 327600.2.a.t

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);
 
\(q - q^{7} - 4 q^{11} - q^{13} - 6 q^{17} - 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

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 327600t

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.t1 327600t1 \([0, 0, 0, -24003615, 45264946750]\) \(65565618540844760336/188428167837\) \(4395652299301536000\) \([2]\) \(15040512\) \(2.8078\) \(\Gamma_0(N)\)-optimal
327600.t2 327600t2 \([0, 0, 0, -23694915, 46485855250]\) \(-15767094823546327124/879851321206767\) \(-82100686484445842304000\) \([2]\) \(30081024\) \(3.1544\)