Properties

Label 212160gm
Number of curves $4$
Conductor $212160$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("gm1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 212160gm

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
212160.cw3 212160gm1 \([0, -1, 0, -3485, 67725]\) \(4572531595264/776953125\) \(795600000000\) \([2]\) \(294912\) \(1.0042\) \(\Gamma_0(N)\)-optimal
212160.cw2 212160gm2 \([0, -1, 0, -15985, -709775]\) \(27572037674704/2472575625\) \(40510679040000\) \([2, 2]\) \(589824\) \(1.3508\)  
212160.cw4 212160gm3 \([0, -1, 0, 18015, -3354975]\) \(9865576607324/79640206425\) \(-5219300568268800\) \([2]\) \(1179648\) \(1.6974\)  
212160.cw1 212160gm4 \([0, -1, 0, -249985, -48024575]\) \(26362547147244676/244298925\) \(16010374348800\) \([2]\) \(1179648\) \(1.6974\)  

Rank

sage: E.rank()
 

The elliptic curves in class 212160gm have rank \(1\).

Complex multiplication

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

Modular form 212160.2.a.gm

sage: E.q_eigenform(10)
 
\(q - q^{3} + q^{5} + q^{9} - 4q^{11} - q^{13} - q^{15} + q^{17} + 8q^{19} + O(q^{20})\)  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 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.