Properties

Label 418950kk
Number of curves $4$
Conductor $418950$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("kk1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 418950kk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
418950.kk3 418950kk1 \([1, -1, 1, -342005, 76607997]\) \(3301293169/22800\) \(30554180606250000\) \([2]\) \(4718592\) \(1.9967\) \(\Gamma_0(N)\)-optimal
418950.kk2 418950kk2 \([1, -1, 1, -562505, -34083003]\) \(14688124849/8122500\) \(10884926840976562500\) \([2, 2]\) \(9437184\) \(2.3433\)  
418950.kk4 418950kk3 \([1, -1, 1, 2193745, -271120503]\) \(871257511151/527800050\) \(-707302546126657031250\) \([2]\) \(18874368\) \(2.6899\)  
418950.kk1 418950kk4 \([1, -1, 1, -6846755, -6883915503]\) \(26487576322129/44531250\) \(59676133996582031250\) \([2]\) \(18874368\) \(2.6899\)  

Rank

sage: E.rank()
 

The elliptic curves in class 418950kk have rank \(1\).

Complex multiplication

The elliptic curves in class 418950kk do not have complex multiplication.

Modular form 418950.2.a.kk

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + q^{8} - 4q^{11} + 2q^{13} + q^{16} - 2q^{17} + q^{19} + O(q^{20})\)  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.