Properties

Label 212160.d
Number of curves $4$
Conductor $212160$
CM no
Rank $2$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 212160.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
212160.d1 212160hd4 \([0, -1, 0, -360961, -62106239]\) \(79364416584061444/20404090514925\) \(1337202475986124800\) \([2]\) \(3145728\) \(2.1874\)  
212160.d2 212160hd2 \([0, -1, 0, -126961, 16658161]\) \(13813960087661776/714574355625\) \(11707586242560000\) \([2, 2]\) \(1572864\) \(1.8409\)  
212160.d3 212160hd1 \([0, -1, 0, -125341, 17121805]\) \(212670222886967296/616241925\) \(631031731200\) \([2]\) \(786432\) \(1.4943\) \(\Gamma_0(N)\)-optimal
212160.d4 212160hd3 \([0, -1, 0, 81119, 65723425]\) \(900753985478876/29018422265625\) \(-1901751321600000000\) \([2]\) \(3145728\) \(2.1874\)  

Rank

sage: E.rank()
 

The elliptic curves in class 212160.d have rank \(2\).

Complex multiplication

The elliptic curves in class 212160.d do not have complex multiplication.

Modular form 212160.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{5} - 4q^{7} + q^{9} - q^{13} + q^{15} + q^{17} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.