Properties

Label 26.2.287...125.1
Degree $26$
Signature $[2, 12]$
Discriminant $2.877\times 10^{36}$
Root discriminant $25.25$
Ramified primes $5, 191$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{26}$ (as 26T3)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 2*x^25 - 4*x^24 + 16*x^23 - 11*x^22 - 29*x^21 + 88*x^20 - 189*x^19 + 510*x^18 - 655*x^17 + 132*x^16 - 179*x^15 + 449*x^14 + 1090*x^13 + 114*x^12 - 781*x^11 - 543*x^10 - 58*x^9 + 245*x^8 + 173*x^7 + 196*x^6 + 159*x^5 + 34*x^4 - 4*x^3 + 6*x^2 - 6*x - 1)
 
gp: K = bnfinit(x^26 - 2*x^25 - 4*x^24 + 16*x^23 - 11*x^22 - 29*x^21 + 88*x^20 - 189*x^19 + 510*x^18 - 655*x^17 + 132*x^16 - 179*x^15 + 449*x^14 + 1090*x^13 + 114*x^12 - 781*x^11 - 543*x^10 - 58*x^9 + 245*x^8 + 173*x^7 + 196*x^6 + 159*x^5 + 34*x^4 - 4*x^3 + 6*x^2 - 6*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -6, 6, -4, 34, 159, 196, 173, 245, -58, -543, -781, 114, 1090, 449, -179, 132, -655, 510, -189, 88, -29, -11, 16, -4, -2, 1]);
 

\( x^{26} - 2 x^{25} - 4 x^{24} + 16 x^{23} - 11 x^{22} - 29 x^{21} + 88 x^{20} - 189 x^{19} + 510 x^{18} - 655 x^{17} + 132 x^{16} - 179 x^{15} + 449 x^{14} + 1090 x^{13} + 114 x^{12} - 781 x^{11} - 543 x^{10} - 58 x^{9} + 245 x^{8} + 173 x^{7} + 196 x^{6} + 159 x^{5} + 34 x^{4} - 4 x^{3} + 6 x^{2} - 6 x - 1 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $26$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 12]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(2877467739962384875567767188720703125\)\(\medspace = 5^{13}\cdot 191^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $25.25$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $5, 191$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{7} a^{19} - \frac{2}{7} a^{18} - \frac{2}{7} a^{17} + \frac{1}{7} a^{16} + \frac{2}{7} a^{15} + \frac{1}{7} a^{14} + \frac{3}{7} a^{13} + \frac{1}{7} a^{12} + \frac{2}{7} a^{11} + \frac{1}{7} a^{10} - \frac{1}{7} a^{9} - \frac{2}{7} a^{8} + \frac{2}{7} a^{7} - \frac{3}{7} a^{6} - \frac{1}{7} a^{5} + \frac{3}{7} a^{3} - \frac{3}{7} a^{2} - \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{20} + \frac{1}{7} a^{18} - \frac{3}{7} a^{17} - \frac{3}{7} a^{16} - \frac{2}{7} a^{15} - \frac{2}{7} a^{14} - \frac{3}{7} a^{12} - \frac{2}{7} a^{11} + \frac{1}{7} a^{10} + \frac{3}{7} a^{9} - \frac{2}{7} a^{8} + \frac{1}{7} a^{7} - \frac{2}{7} a^{5} + \frac{3}{7} a^{4} + \frac{3}{7} a^{3} - \frac{2}{7} a^{2} + \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{21} - \frac{1}{7} a^{18} - \frac{1}{7} a^{17} - \frac{3}{7} a^{16} + \frac{3}{7} a^{15} - \frac{1}{7} a^{14} + \frac{1}{7} a^{13} - \frac{3}{7} a^{12} - \frac{1}{7} a^{11} + \frac{2}{7} a^{10} - \frac{1}{7} a^{9} + \frac{3}{7} a^{8} - \frac{2}{7} a^{7} + \frac{1}{7} a^{6} - \frac{3}{7} a^{5} + \frac{3}{7} a^{4} + \frac{2}{7} a^{3} - \frac{2}{7} a^{2} - \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{22} - \frac{3}{7} a^{18} + \frac{2}{7} a^{17} - \frac{3}{7} a^{16} + \frac{1}{7} a^{15} + \frac{2}{7} a^{14} - \frac{3}{7} a^{11} + \frac{2}{7} a^{9} + \frac{3}{7} a^{8} + \frac{3}{7} a^{7} + \frac{1}{7} a^{6} + \frac{2}{7} a^{5} + \frac{2}{7} a^{4} + \frac{1}{7} a^{3} + \frac{2}{7} a^{2} + \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{287} a^{23} - \frac{13}{287} a^{22} - \frac{19}{287} a^{21} - \frac{1}{287} a^{20} + \frac{6}{287} a^{19} - \frac{36}{287} a^{18} + \frac{94}{287} a^{17} - \frac{31}{287} a^{16} - \frac{104}{287} a^{15} - \frac{38}{287} a^{14} + \frac{127}{287} a^{13} + \frac{17}{287} a^{12} - \frac{76}{287} a^{11} + \frac{6}{41} a^{10} + \frac{124}{287} a^{9} - \frac{116}{287} a^{8} + \frac{17}{287} a^{7} - \frac{64}{287} a^{6} + \frac{138}{287} a^{5} - \frac{22}{287} a^{4} - \frac{4}{287} a^{3} - \frac{17}{287} a^{2} + \frac{132}{287} a - \frac{64}{287}$, $\frac{1}{3157} a^{24} - \frac{5}{3157} a^{23} + \frac{1}{77} a^{22} + \frac{216}{3157} a^{21} - \frac{166}{3157} a^{20} + \frac{31}{451} a^{19} - \frac{1055}{3157} a^{18} + \frac{1049}{3157} a^{17} - \frac{114}{287} a^{16} + \frac{565}{3157} a^{15} + \frac{168}{451} a^{14} + \frac{582}{3157} a^{13} + \frac{32}{451} a^{12} - \frac{976}{3157} a^{11} + \frac{95}{451} a^{10} + \frac{999}{3157} a^{9} - \frac{255}{3157} a^{8} + \frac{933}{3157} a^{7} - \frac{1030}{3157} a^{6} - \frac{1296}{3157} a^{5} + \frac{27}{451} a^{4} - \frac{172}{3157} a^{3} - \frac{127}{3157} a^{2} + \frac{90}{3157} a - \frac{120}{451}$, $\frac{1}{727521618708748885045977562501969} a^{25} - \frac{49442651239952986463251627119}{727521618708748885045977562501969} a^{24} + \frac{78959742694766641589128710489}{66138328973522625913270687500179} a^{23} - \frac{1621240456759326735446183832605}{66138328973522625913270687500179} a^{22} + \frac{3413959274262973020860933171996}{103931659815535555006568223214567} a^{21} + \frac{23392711773189392219013811874614}{727521618708748885045977562501969} a^{20} + \frac{21712716120471879651359878014353}{727521618708748885045977562501969} a^{19} + \frac{38596052982471969214779872111623}{727521618708748885045977562501969} a^{18} + \frac{207022835902018076144151721722786}{727521618708748885045977562501969} a^{17} + \frac{353112347642024470747180564688996}{727521618708748885045977562501969} a^{16} - \frac{10207195801420405595758377712761}{103931659815535555006568223214567} a^{15} - \frac{129725951889539347090625600990251}{727521618708748885045977562501969} a^{14} + \frac{191151991239351518124853320495882}{727521618708748885045977562501969} a^{13} - \frac{206864814899710781905502701754580}{727521618708748885045977562501969} a^{12} + \frac{125577257881677378815166201429995}{727521618708748885045977562501969} a^{11} - \frac{62823569955952138236092220256912}{727521618708748885045977562501969} a^{10} - \frac{35242935786805770553845378029298}{103931659815535555006568223214567} a^{9} - \frac{64506460193649126538370239093463}{727521618708748885045977562501969} a^{8} - \frac{4870844825434003125753391223851}{727521618708748885045977562501969} a^{7} + \frac{199359277577306106991210229568680}{727521618708748885045977562501969} a^{6} + \frac{139836313359051538340019836351031}{727521618708748885045977562501969} a^{5} + \frac{179214424194959065886929379479390}{727521618708748885045977562501969} a^{4} - \frac{344542017870821083552315359045481}{727521618708748885045977562501969} a^{3} - \frac{248080363447413445403298381957585}{727521618708748885045977562501969} a^{2} - \frac{126338769065862293447375040576748}{727521618708748885045977562501969} a + \frac{46580161606323284598004571742589}{103931659815535555006568223214567}$

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $13$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 62790605.5500462 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{12}\cdot 62790605.5500462 \cdot 1}{2\sqrt{2877467739962384875567767188720703125}}\approx 0.280270801626651$ (assuming GRH)

Galois group

$D_{26}$ (as 26T3):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 52
The 16 conjugacy class representatives for $D_{26}$
Character table for $D_{26}$

Intermediate fields

\(\Q(\sqrt{5}) \), 13.1.48551226272641.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 26 sibling: 26.0.549596338332815511233443533045654296875.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $26$ $26$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $26$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $26$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/59.13.0.1}{13} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
5Data not computed
$191$$\Q_{191}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{191}$$x + 2$$1$$1$$0$Trivial$[\ ]$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.955.2t1.a.a$1$ $ 5 \cdot 191 $ \(\Q(\sqrt{-955}) \) $C_2$ (as 2T1) $1$ $-1$
1.191.2t1.a.a$1$ $ 191 $ \(\Q(\sqrt{-191}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
* 2.4775.26t3.b.c$2$ $ 5^{2} \cdot 191 $ 26.2.2877467739962384875567767188720703125.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.191.13t2.a.c$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.b$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.4775.26t3.b.f$2$ $ 5^{2} \cdot 191 $ 26.2.2877467739962384875567767188720703125.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.4775.26t3.b.b$2$ $ 5^{2} \cdot 191 $ 26.2.2877467739962384875567767188720703125.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.191.13t2.a.a$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.d$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.f$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.4775.26t3.b.d$2$ $ 5^{2} \cdot 191 $ 26.2.2877467739962384875567767188720703125.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.4775.26t3.b.a$2$ $ 5^{2} \cdot 191 $ 26.2.2877467739962384875567767188720703125.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.191.13t2.a.e$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.4775.26t3.b.e$2$ $ 5^{2} \cdot 191 $ 26.2.2877467739962384875567767188720703125.1 $D_{26}$ (as 26T3) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.