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Properties

Label 51150.n
Number of curves $4$
Conductor $51150$
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([1, 1, 0, -218296875, -1238913937875]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 0, -218296875, -1238913937875]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 0, -218296875, -1238913937875]) 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 51150.n have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 + T\)
\(5\)\(1\)
\(11\)\(1 - T\)
\(31\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 - 2 T + 7 T^{2}\) 1.7.ac
\(13\) \( 1 - 6 T + 13 T^{2}\) 1.13.ag
\(17\) \( 1 + 8 T + 17 T^{2}\) 1.17.i
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 + 4 T + 23 T^{2}\) 1.23.e
\(29\) \( 1 + 10 T + 29 T^{2}\) 1.29.k
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 51150.n do not have complex multiplication.

Modular form 51150.2.a.n

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);
 
\(q - q^{2} - q^{3} + q^{4} + q^{6} + 2 q^{7} - q^{8} + q^{9} + q^{11} - q^{12} + 6 q^{13} - 2 q^{14} + q^{16} - 8 q^{17} - q^{18} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

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 51150.n

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
51150.n1 51150k3 \([1, 1, 0, -218296875, -1238913937875]\) \(73628549562506871957390001/178215946908754500240\) \(2784624170449289066250000\) \([2]\) \(21120000\) \(3.5693\)  
51150.n2 51150k4 \([1, 1, 0, -137771375, -2164393509375]\) \(-18508902577171306222471921/118801759721890483665900\) \(-1856277495654538807279687500\) \([2]\) \(42240000\) \(3.9159\)  
51150.n3 51150k1 \([1, 1, 0, -12166875, 16326532125]\) \(12747965531857798561201/2986780262400000\) \(46668441600000000000\) \([2]\) \(4224000\) \(2.7646\) \(\Gamma_0(N)\)-optimal
51150.n4 51150k2 \([1, 1, 0, -10758875, 20250628125]\) \(-8814635019030000319921/6242069790000000000\) \(-97532340468750000000000\) \([2]\) \(8448000\) \(3.1111\)