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Properties

Label 121680.eo
Number of curves $6$
Conductor $121680$
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, -3701506107, 86679384534106]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -3701506107, 86679384534106]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -3701506107, 86679384534106]) 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 121680.eo have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(5\)\(1 - T\)
\(13\)\(1\)
 
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.ae
\(17\) \( 1 + 2 T + 17 T^{2}\) 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.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 121680.eo do not have complex multiplication.

Modular form 121680.2.a.eo

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

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

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 121680.eo

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
121680.eo1 121680bg4 \([0, 0, 0, -3701506107, 86679384534106]\) \(1556580279686303289604/114075\) \(411034311204940800\) \([2]\) \(41287680\) \(3.7494\)  
121680.eo2 121680bg6 \([0, 0, 0, -813066267, -7450293588374]\) \(8248670337458940482/1446075439453125\) \(10420979569686562500000000000\) \([2]\) \(82575360\) \(4.0959\)  
121680.eo3 121680bg3 \([0, 0, 0, -236485587, 1291015468834]\) \(405929061432816484/35083409765625\) \(126412317928857027600000000\) \([2, 2]\) \(41287680\) \(3.7494\)  
121680.eo4 121680bg2 \([0, 0, 0, -231344607, 1354359539806]\) \(1520107298839022416/13013105625\) \(11722184762675905440000\) \([2, 2]\) \(20643840\) \(3.4028\)  
121680.eo5 121680bg1 \([0, 0, 0, -14138202, 22145775379]\) \(-5551350318708736/550618236675\) \(-30999751759887642154800\) \([2]\) \(10321920\) \(3.0562\) \(\Gamma_0(N)\)-optimal
121680.eo6 121680bg5 \([0, 0, 0, 257839413, 5978303983834]\) \(263059523447441758/2294739983908125\) \(-16536784899059620333482240000\) \([2]\) \(82575360\) \(4.0959\)