Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 1, 0, -3200, -70752]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 0, -3200, -70752]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 0, -3200, -70752]) 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 120.b have rank \(0\).

L-function data

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

Complex multiplication

The elliptic curves in class 120.b do not have complex multiplication.

Modular form 120.2.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);
 
\(q + q^{3} + q^{5} + q^{9} - 4 q^{11} + 6 q^{13} + q^{15} - 6 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

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 120.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
120.b1 120a5 \([0, 1, 0, -3200, -70752]\) \(1770025017602/75\) \(153600\) \([2]\) \(64\) \(0.48000\)  
120.b2 120a3 \([0, 1, 0, -200, -1152]\) \(868327204/5625\) \(5760000\) \([2, 2]\) \(32\) \(0.13343\)  
120.b3 120a6 \([0, 1, 0, -80, -2400]\) \(-27995042/1171875\) \(-2400000000\) \([2]\) \(64\) \(0.48000\)  
120.b4 120a2 \([0, 1, 0, -20, 0]\) \(3631696/2025\) \(518400\) \([2, 4]\) \(16\) \(-0.21315\)  
120.b5 120a1 \([0, 1, 0, -15, 18]\) \(24918016/45\) \(720\) \([4]\) \(8\) \(-0.55972\) \(\Gamma_0(N)\)-optimal
120.b6 120a4 \([0, 1, 0, 80, 80]\) \(54607676/32805\) \(-33592320\) \([4]\) \(32\) \(0.13343\)