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Properties

Label 366912.nt
Number of curves $4$
Conductor $366912$
CM no
Rank $0$
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, -1712844, 862828848]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -1712844, 862828848]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -1712844, 862828848]) 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 366912.nt have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(7\)\(1\)
\(13\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 4 T + 23 T^{2}\) 1.23.e
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 366912.nt do not have complex multiplication.

Modular form 366912.2.a.nt

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);
 
\(q + 2 q^{5} + 4 q^{11} + q^{13} + 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

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 366912.nt

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
366912.nt1 366912nt3 \([0, 0, 0, -1712844, 862828848]\) \(197747699976/91\) \(255744967016448\) \([2]\) \(3538944\) \(2.1033\)  
366912.nt2 366912nt4 \([0, 0, 0, -231084, -22967280]\) \(485587656/199927\) \(561871692535136256\) \([2]\) \(3538944\) \(2.1033\)  
366912.nt3 366912nt2 \([0, 0, 0, -107604, 13335840]\) \(392223168/8281\) \(2909098999812096\) \([2, 2]\) \(1769472\) \(1.7567\)  
366912.nt4 366912nt1 \([0, 0, 0, 441, 629748]\) \(1728/31213\) \(-171329147825472\) \([2]\) \(884736\) \(1.4101\) \(\Gamma_0(N)\)-optimal