Properties

Label 123840.dk
Number of curves $4$
Conductor $123840$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("dk1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 123840.dk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
123840.dk1 123840cd4 \([0, 0, 0, -482988, -128316112]\) \(65202655558249/512820150\) \(98001456817766400\) \([2]\) \(1179648\) \(2.0885\)  
123840.dk2 123840cd2 \([0, 0, 0, -50988, 1111088]\) \(76711450249/41602500\) \(7950361559040000\) \([2, 2]\) \(589824\) \(1.7419\)  
123840.dk3 123840cd1 \([0, 0, 0, -39468, 3014192]\) \(35578826569/51600\) \(9860913561600\) \([2]\) \(294912\) \(1.3954\) \(\Gamma_0(N)\)-optimal
123840.dk4 123840cd3 \([0, 0, 0, 196692, 8739632]\) \(4403686064471/2721093750\) \(-520009113600000000\) \([2]\) \(1179648\) \(2.0885\)  

Rank

sage: E.rank()
 

The elliptic curves in class 123840.dk have rank \(1\).

Complex multiplication

The elliptic curves in class 123840.dk do not have complex multiplication.

Modular form 123840.2.a.dk

sage: E.q_eigenform(10)
 
\(q - q^{5} + 4q^{7} + 2q^{13} - 2q^{17} - 4q^{19} + O(q^{20})\)  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 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.