Properties

Label 121680.er
Number of curves $6$
Conductor $121680$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("er1")
 
E.isogeny_class()
 

Elliptic curves in class 121680.er

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
121680.er1 121680bh6 \([0, 0, 0, -4867707, -4133659894]\) \(1770025017602/75\) \(540479041689600\) \([2]\) \(1966080\) \(2.3118\)  
121680.er2 121680bh4 \([0, 0, 0, -304707, -64376494]\) \(868327204/5625\) \(20267964063360000\) \([2, 2]\) \(983040\) \(1.9652\)  
121680.er3 121680bh5 \([0, 0, 0, -122187, -140779366]\) \(-27995042/1171875\) \(-8444985026400000000\) \([2]\) \(1966080\) \(2.3118\)  
121680.er4 121680bh2 \([0, 0, 0, -30927, 399854]\) \(3631696/2025\) \(1824116765702400\) \([2, 2]\) \(491520\) \(1.6186\)  
121680.er5 121680bh1 \([0, 0, 0, -23322, 1368731]\) \(24918016/45\) \(2533495507920\) \([2]\) \(245760\) \(1.2721\) \(\Gamma_0(N)\)-optimal
121680.er6 121680bh3 \([0, 0, 0, 121173, 3168074]\) \(54607676/32805\) \(-118202766417515520\) \([2]\) \(983040\) \(1.9652\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 121680.2.a.er

sage: E.q_eigenform(10)
 
\(q + q^{5} + 4 q^{11} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.