Properties

Label 2112r
Number of curves $4$
Conductor $2112$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("r1")
 
E.isogeny_class()
 

Elliptic curves in class 2112r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2112.r4 2112r1 \([0, 1, 0, -129, -33]\) \(912673/528\) \(138412032\) \([2]\) \(768\) \(0.25162\) \(\Gamma_0(N)\)-optimal
2112.r2 2112r2 \([0, 1, 0, -1409, -20769]\) \(1180932193/4356\) \(1141899264\) \([2, 2]\) \(1536\) \(0.59819\)  
2112.r1 2112r3 \([0, 1, 0, -22529, -1309089]\) \(4824238966273/66\) \(17301504\) \([2]\) \(3072\) \(0.94477\)  
2112.r3 2112r4 \([0, 1, 0, -769, -39073]\) \(-192100033/2371842\) \(-621764149248\) \([4]\) \(3072\) \(0.94477\)  

Rank

sage: E.rank()
 

The elliptic curves in class 2112r have rank \(1\).

Complex multiplication

The elliptic curves in class 2112r do not have complex multiplication.

Modular form 2112.2.a.r

sage: E.q_eigenform(10)
 
\(q + q^{3} - 2 q^{5} - 4 q^{7} + q^{9} + q^{11} + 6 q^{13} - 2 q^{15} + 2 q^{17} - 4 q^{19} + O(q^{20})\) 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 Cremona 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 Cremona labels.