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Properties

Label 120984.b
Number of curves $6$
Conductor $120984$
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, -1937424, 1038614220]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -1937424, 1038614220]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -1937424, 1038614220]) 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 120984.b have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(71\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 120984.b do not have complex multiplication.

Modular form 120984.2.a.b

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

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

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 120984.b

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
120984.b1 120984h6 \([0, -1, 0, -1937424, 1038614220]\) \(3065617154/9\) \(2361144433231872\) \([2]\) \(1433600\) \(2.1791\)  
120984.b2 120984h4 \([0, -1, 0, -324304, -70970372]\) \(28756228/3\) \(393524072205312\) \([2]\) \(716800\) \(1.8326\)  
120984.b3 120984h3 \([0, -1, 0, -122664, 15815484]\) \(1556068/81\) \(10625149949543424\) \([2, 2]\) \(716800\) \(1.8326\)  
120984.b4 120984h2 \([0, -1, 0, -21844, -920636]\) \(35152/9\) \(295143054153984\) \([2, 2]\) \(358400\) \(1.4860\)  
120984.b5 120984h1 \([0, -1, 0, 3361, -93912]\) \(2048/3\) \(-6148813628208\) \([2]\) \(179200\) \(1.1394\) \(\Gamma_0(N)\)-optimal
120984.b6 120984h5 \([0, -1, 0, 78976, 62515308]\) \(207646/6561\) \(-1721274291826034688\) \([2]\) \(1433600\) \(2.1791\)