Properties

Label 132600ck
Number of curves $6$
Conductor $132600$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("ck1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 132600ck

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
132600.p4 132600ck1 \([0, -1, 0, -414383, 102810012]\) \(31476797652269056/49725\) \(12431250000\) \([2]\) \(540672\) \(1.6314\) \(\Gamma_0(N)\)-optimal
132600.p3 132600ck2 \([0, -1, 0, -414508, 102745012]\) \(1969080716416336/2472575625\) \(9890302500000000\) \([2, 2]\) \(1081344\) \(1.9779\)  
132600.p5 132600ck3 \([0, -1, 0, -304008, 158658012]\) \(-194204905090564/566398828125\) \(-9062381250000000000\) \([2]\) \(2162688\) \(2.3245\)  
132600.p2 132600ck4 \([0, -1, 0, -527008, 42670012]\) \(1011710313226084/536724738225\) \(8587595811600000000\) \([2, 2]\) \(2162688\) \(2.3245\)  
132600.p6 132600ck5 \([0, -1, 0, 2007992, 331660012]\) \(27980756504588158/17683545112935\) \(-565873443613920000000\) \([2]\) \(4325376\) \(2.6711\)  
132600.p1 132600ck6 \([0, -1, 0, -4862008, -4092919988]\) \(397210600760070242/3536192675535\) \(113158165617120000000\) \([2]\) \(4325376\) \(2.6711\)  

Rank

sage: E.rank()
 

The elliptic curves in class 132600ck have rank \(1\).

Complex multiplication

The elliptic curves in class 132600ck do not have complex multiplication.

Modular form 132600.2.a.ck

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

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.