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Properties

Label 26.2.955...125.1
Degree $26$
Signature $[2, 12]$
Discriminant $9.559\times 10^{41}$
Root discriminant $41.17$
Ramified primes $5, 19, 29$
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 - 4*x^25 - x^24 + 12*x^23 - 20*x^22 - 91*x^21 + 245*x^20 + 945*x^19 + 237*x^18 - 2538*x^17 - 3465*x^16 + 4625*x^15 + 16764*x^14 + 17173*x^13 + 6911*x^12 - 948*x^11 + 1857*x^10 + 6384*x^9 + 2100*x^8 - 1050*x^7 + 280*x^6 + 490*x^5 - 352*x^4 - 172*x^3 - 11*x^2 + 8*x - 1)
 
gp: K = bnfinit(x^26 - 4*x^25 - x^24 + 12*x^23 - 20*x^22 - 91*x^21 + 245*x^20 + 945*x^19 + 237*x^18 - 2538*x^17 - 3465*x^16 + 4625*x^15 + 16764*x^14 + 17173*x^13 + 6911*x^12 - 948*x^11 + 1857*x^10 + 6384*x^9 + 2100*x^8 - 1050*x^7 + 280*x^6 + 490*x^5 - 352*x^4 - 172*x^3 - 11*x^2 + 8*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 8, -11, -172, -352, 490, 280, -1050, 2100, 6384, 1857, -948, 6911, 17173, 16764, 4625, -3465, -2538, 237, 945, 245, -91, -20, 12, -1, -4, 1]);
 

\( x^{26} - 4 x^{25} - x^{24} + 12 x^{23} - 20 x^{22} - 91 x^{21} + 245 x^{20} + 945 x^{19} + 237 x^{18} - 2538 x^{17} - 3465 x^{16} + 4625 x^{15} + 16764 x^{14} + 17173 x^{13} + 6911 x^{12} - 948 x^{11} + 1857 x^{10} + 6384 x^{9} + 2100 x^{8} - 1050 x^{7} + 280 x^{6} + 490 x^{5} - 352 x^{4} - 172 x^{3} - 11 x^{2} + 8 x - 1 \)

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:  \(955936936346220716198217427028321533203125\)\(\medspace = 5^{13}\cdot 19^{12}\cdot 29^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $41.17$
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:  $5, 19, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{19} a^{12} + \frac{8}{19} a^{11} - \frac{6}{19} a^{10} - \frac{8}{19} a^{9} + \frac{3}{19} a^{8} - \frac{3}{19} a^{7} - \frac{4}{19} a^{6} - \frac{8}{19} a^{5} + \frac{5}{19} a^{4} + \frac{1}{19} a^{3} + \frac{4}{19} a^{2} - \frac{5}{19} a + \frac{3}{19}$, $\frac{1}{19} a^{13} + \frac{6}{19} a^{11} + \frac{2}{19} a^{10} - \frac{9}{19} a^{9} - \frac{8}{19} a^{8} + \frac{1}{19} a^{7} + \frac{5}{19} a^{6} - \frac{7}{19} a^{5} - \frac{1}{19} a^{4} - \frac{4}{19} a^{3} + \frac{1}{19} a^{2} + \frac{5}{19} a - \frac{5}{19}$, $\frac{1}{19} a^{14} - \frac{8}{19} a^{11} + \frac{8}{19} a^{10} + \frac{2}{19} a^{9} + \frac{2}{19} a^{8} + \frac{4}{19} a^{7} - \frac{2}{19} a^{6} + \frac{9}{19} a^{5} + \frac{4}{19} a^{4} - \frac{5}{19} a^{3} + \frac{6}{19} a + \frac{1}{19}$, $\frac{1}{19} a^{15} - \frac{4}{19} a^{11} - \frac{8}{19} a^{10} - \frac{5}{19} a^{9} + \frac{9}{19} a^{8} - \frac{7}{19} a^{7} - \frac{4}{19} a^{6} - \frac{3}{19} a^{5} - \frac{3}{19} a^{4} + \frac{8}{19} a^{3} - \frac{1}{19} a + \frac{5}{19}$, $\frac{1}{19} a^{16} + \frac{5}{19} a^{11} + \frac{9}{19} a^{10} - \frac{4}{19} a^{9} + \frac{5}{19} a^{8} + \frac{3}{19} a^{7} + \frac{3}{19} a^{5} + \frac{9}{19} a^{4} + \frac{4}{19} a^{3} - \frac{4}{19} a^{2} + \frac{4}{19} a - \frac{7}{19}$, $\frac{1}{19} a^{17} + \frac{7}{19} a^{11} + \frac{7}{19} a^{10} + \frac{7}{19} a^{9} + \frac{7}{19} a^{8} - \frac{4}{19} a^{7} + \frac{4}{19} a^{6} - \frac{8}{19} a^{5} - \frac{2}{19} a^{4} - \frac{9}{19} a^{3} + \frac{3}{19} a^{2} - \frac{1}{19} a + \frac{4}{19}$, $\frac{1}{19} a^{18} + \frac{8}{19} a^{11} - \frac{8}{19} a^{10} + \frac{6}{19} a^{9} - \frac{6}{19} a^{8} + \frac{6}{19} a^{7} + \frac{1}{19} a^{6} - \frac{3}{19} a^{5} - \frac{6}{19} a^{4} - \frac{4}{19} a^{3} + \frac{9}{19} a^{2} + \frac{1}{19} a - \frac{2}{19}$, $\frac{1}{19} a^{19} + \frac{4}{19} a^{11} - \frac{3}{19} a^{10} + \frac{1}{19} a^{9} + \frac{1}{19} a^{8} + \frac{6}{19} a^{7} - \frac{9}{19} a^{6} + \frac{1}{19} a^{5} - \frac{6}{19} a^{4} + \frac{1}{19} a^{3} + \frac{7}{19} a^{2} - \frac{5}{19}$, $\frac{1}{19} a^{20} + \frac{3}{19} a^{11} + \frac{6}{19} a^{10} - \frac{5}{19} a^{9} - \frac{6}{19} a^{8} + \frac{3}{19} a^{7} - \frac{2}{19} a^{6} + \frac{7}{19} a^{5} + \frac{3}{19} a^{3} + \frac{3}{19} a^{2} - \frac{4}{19} a + \frac{7}{19}$, $\frac{1}{19} a^{21} + \frac{1}{19} a^{11} - \frac{6}{19} a^{10} - \frac{1}{19} a^{9} - \frac{6}{19} a^{8} + \frac{7}{19} a^{7} + \frac{5}{19} a^{5} + \frac{7}{19} a^{4} + \frac{3}{19} a^{2} + \frac{3}{19} a - \frac{9}{19}$, $\frac{1}{361} a^{22} - \frac{6}{361} a^{21} + \frac{6}{361} a^{20} + \frac{7}{361} a^{19} + \frac{3}{361} a^{18} + \frac{4}{361} a^{17} - \frac{4}{361} a^{16} + \frac{6}{361} a^{15} - \frac{3}{361} a^{14} + \frac{1}{361} a^{13} + \frac{83}{361} a^{11} + \frac{11}{361} a^{10} + \frac{78}{361} a^{9} + \frac{117}{361} a^{8} - \frac{175}{361} a^{7} + \frac{35}{361} a^{6} + \frac{119}{361} a^{5} + \frac{65}{361} a^{4} + \frac{98}{361} a^{3} - \frac{48}{361} a^{2} + \frac{89}{361} a + \frac{175}{361}$, $\frac{1}{361} a^{23} + \frac{8}{361} a^{21} + \frac{5}{361} a^{20} + \frac{7}{361} a^{19} + \frac{3}{361} a^{18} + \frac{1}{361} a^{17} + \frac{1}{361} a^{16} - \frac{5}{361} a^{15} + \frac{2}{361} a^{14} + \frac{6}{361} a^{13} + \frac{7}{361} a^{12} - \frac{156}{361} a^{11} - \frac{179}{361} a^{10} + \frac{129}{361} a^{9} + \frac{33}{361} a^{8} - \frac{141}{361} a^{7} - \frac{13}{361} a^{6} + \frac{1}{19} a^{5} + \frac{32}{361} a^{4} - \frac{125}{361} a^{3} + \frac{10}{361} a^{2} + \frac{139}{361} a + \frac{62}{361}$, $\frac{1}{10469} a^{24} + \frac{5}{10469} a^{23} + \frac{2}{10469} a^{22} + \frac{81}{10469} a^{21} - \frac{42}{10469} a^{20} + \frac{15}{10469} a^{19} - \frac{2}{10469} a^{18} + \frac{210}{10469} a^{17} - \frac{109}{10469} a^{16} + \frac{131}{10469} a^{15} + \frac{262}{10469} a^{14} + \frac{31}{10469} a^{13} - \frac{254}{10469} a^{12} - \frac{2046}{10469} a^{11} - \frac{2865}{10469} a^{10} + \frac{590}{10469} a^{9} - \frac{3148}{10469} a^{8} - \frac{1720}{10469} a^{7} + \frac{3126}{10469} a^{6} + \frac{2377}{10469} a^{5} - \frac{1362}{10469} a^{4} - \frac{3635}{10469} a^{3} - \frac{3152}{10469} a^{2} - \frac{2513}{10469} a + \frac{3326}{10469}$, $\frac{1}{46918419213254345422166367919} a^{25} + \frac{1527042082391566083026585}{46918419213254345422166367919} a^{24} - \frac{53951787880502506107663727}{46918419213254345422166367919} a^{23} + \frac{148161123126790968504673}{998264238579879689833326977} a^{22} - \frac{291381970624366155526344756}{46918419213254345422166367919} a^{21} + \frac{1127073959911118586065344312}{46918419213254345422166367919} a^{20} - \frac{842590638542166067556200086}{46918419213254345422166367919} a^{19} - \frac{1032419895275949202209170799}{46918419213254345422166367919} a^{18} + \frac{190539840533213623705008591}{46918419213254345422166367919} a^{17} + \frac{461915262856974295240903714}{46918419213254345422166367919} a^{16} - \frac{897551086096097836781372705}{46918419213254345422166367919} a^{15} - \frac{162565543595442424721496478}{46918419213254345422166367919} a^{14} + \frac{41290670128908540524480749}{2759907012544373260127433407} a^{13} + \frac{1111613048963195534719501388}{46918419213254345422166367919} a^{12} - \frac{11969770065629851630799249635}{46918419213254345422166367919} a^{11} + \frac{359227320574783710509762066}{3609109170250334263243566763} a^{10} - \frac{4449545675296070812974027896}{46918419213254345422166367919} a^{9} - \frac{7045097134959957303631693348}{46918419213254345422166367919} a^{8} + \frac{517136809618361944690324034}{1617876524594977428350564411} a^{7} + \frac{10827172117050790505976744528}{46918419213254345422166367919} a^{6} + \frac{13445269739731626048377339395}{46918419213254345422166367919} a^{5} + \frac{1036788429123464148906714799}{3609109170250334263243566763} a^{4} - \frac{13821514445497216003902292470}{46918419213254345422166367919} a^{3} + \frac{9810428790835055330665922205}{46918419213254345422166367919} a^{2} + \frac{10981164631383031143555429489}{46918419213254345422166367919} a + \frac{7632933710464288095085546554}{46918419213254345422166367919}$

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:  \( 74302156406.7933 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{12}\cdot 74302156406.7933 \cdot 1}{2\sqrt{955936936346220716198217427028321533203125}}\approx 0.575407095196568$ (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

\(\Q(\sqrt{5}) \), 13.1.27983987175790801.1

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

Sibling fields

Degree 26 sibling: 26.0.526721251926767614625217802292605164794921875.1

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 $26$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{13}$ R $26$ R ${\href{/LocalNumberField/31.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/41.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.2755.2t1.a.a$1$ $ 5 \cdot 19 \cdot 29 $ \(\Q(\sqrt{-2755}) \) $C_2$ (as 2T1) $1$ $-1$
1.551.2t1.a.a$1$ $ 19 \cdot 29 $ \(\Q(\sqrt{-551}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
* 2.13775.26t3.b.d$2$ $ 5^{2} \cdot 19 \cdot 29 $ 26.2.955936936346220716198217427028321533203125.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.551.13t2.a.e$2$ $ 19 \cdot 29 $ 13.1.27983987175790801.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.551.13t2.a.c$2$ $ 19 \cdot 29 $ 13.1.27983987175790801.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.13775.26t3.b.b$2$ $ 5^{2} \cdot 19 \cdot 29 $ 26.2.955936936346220716198217427028321533203125.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.551.13t2.a.d$2$ $ 19 \cdot 29 $ 13.1.27983987175790801.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.13775.26t3.b.a$2$ $ 5^{2} \cdot 19 \cdot 29 $ 26.2.955936936346220716198217427028321533203125.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.551.13t2.a.a$2$ $ 19 \cdot 29 $ 13.1.27983987175790801.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.551.13t2.a.b$2$ $ 19 \cdot 29 $ 13.1.27983987175790801.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.551.13t2.a.f$2$ $ 19 \cdot 29 $ 13.1.27983987175790801.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.13775.26t3.b.e$2$ $ 5^{2} \cdot 19 \cdot 29 $ 26.2.955936936346220716198217427028321533203125.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.13775.26t3.b.c$2$ $ 5^{2} \cdot 19 \cdot 29 $ 26.2.955936936346220716198217427028321533203125.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.13775.26t3.b.f$2$ $ 5^{2} \cdot 19 \cdot 29 $ 26.2.955936936346220716198217427028321533203125.1 $D_{26}$ (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 *.