Properties

Label 121680.e
Number of curves $8$
Conductor $121680$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 121680.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
121680.e1 121680dz7 \([0, 0, 0, -129796563, -569171178862]\) \(16778985534208729/81000\) \(1167434730049536000\) \([2]\) \(10616832\) \(3.0891\)  
121680.e2 121680dz8 \([0, 0, 0, -11036883, -1920823918]\) \(10316097499609/5859375000\) \(84449850264000000000000\) \([2]\) \(10616832\) \(3.0891\)  
121680.e3 121680dz6 \([0, 0, 0, -8116563, -8883450862]\) \(4102915888729/9000000\) \(129714970005504000000\) \([2, 2]\) \(5308416\) \(2.7425\)  
121680.e4 121680dz5 \([0, 0, 0, -7021443, 7161152258]\) \(2656166199049/33750\) \(486431137520640000\) \([2]\) \(3538944\) \(2.5398\)  
121680.e5 121680dz4 \([0, 0, 0, -1667523, -713879998]\) \(35578826569/5314410\) \(76595392638550056960\) \([2]\) \(3538944\) \(2.5398\)  
121680.e6 121680dz2 \([0, 0, 0, -450723, 105513122]\) \(702595369/72900\) \(1050691257044582400\) \([2, 2]\) \(1769472\) \(2.1932\)  
121680.e7 121680dz3 \([0, 0, 0, -329043, -237746158]\) \(-273359449/1536000\) \(-22138021547606016000\) \([2]\) \(2654208\) \(2.3959\)  
121680.e8 121680dz1 \([0, 0, 0, 35997, 8071778]\) \(357911/2160\) \(-31131592801320960\) \([2]\) \(884736\) \(1.8466\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 121680.e have rank \(1\).

Complex multiplication

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

Modular form 121680.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{5} - 4 q^{7} - 6 q^{17} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.