Properties

Label 26.2.486...125.1
Degree $26$
Signature $[2, 12]$
Discriminant $4.864\times 10^{42}$
Root discriminant $43.83$
Ramified primes $5, 631$
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 + 20*x^24 - 36*x^23 + 151*x^22 - 166*x^21 + 575*x^20 - 1143*x^19 + 1160*x^18 - 4286*x^17 + 2641*x^16 - 4158*x^15 + 10417*x^14 + 3815*x^13 + 21287*x^12 + 17505*x^11 + 19313*x^10 + 20293*x^9 + 10822*x^8 + 10971*x^7 + 6883*x^6 + 3740*x^5 + 2969*x^4 + 837*x^3 + 388*x^2 + 17*x - 1)
 
gp: K = bnfinit(x^26 - 4*x^25 + 20*x^24 - 36*x^23 + 151*x^22 - 166*x^21 + 575*x^20 - 1143*x^19 + 1160*x^18 - 4286*x^17 + 2641*x^16 - 4158*x^15 + 10417*x^14 + 3815*x^13 + 21287*x^12 + 17505*x^11 + 19313*x^10 + 20293*x^9 + 10822*x^8 + 10971*x^7 + 6883*x^6 + 3740*x^5 + 2969*x^4 + 837*x^3 + 388*x^2 + 17*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 17, 388, 837, 2969, 3740, 6883, 10971, 10822, 20293, 19313, 17505, 21287, 3815, 10417, -4158, 2641, -4286, 1160, -1143, 575, -166, 151, -36, 20, -4, 1]);
 

\( x^{26} - 4 x^{25} + 20 x^{24} - 36 x^{23} + 151 x^{22} - 166 x^{21} + 575 x^{20} - 1143 x^{19} + 1160 x^{18} - 4286 x^{17} + 2641 x^{16} - 4158 x^{15} + 10417 x^{14} + 3815 x^{13} + 21287 x^{12} + 17505 x^{11} + 19313 x^{10} + 20293 x^{9} + 10822 x^{8} + 10971 x^{7} + 6883 x^{6} + 3740 x^{5} + 2969 x^{4} + 837 x^{3} + 388 x^{2} + 17 x - 1 \)

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:  \(4863650591176530100056566961783399658203125\)\(\medspace = 5^{13}\cdot 631^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $43.83$
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:  $5, 631$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{7}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{8} - \frac{2}{9}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{9} - \frac{2}{9} a$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{10} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{19} + \frac{1}{9} a^{11} - \frac{2}{9} a^{3}$, $\frac{1}{27} a^{20} - \frac{1}{27} a^{16} - \frac{1}{9} a^{15} - \frac{1}{9} a^{13} + \frac{4}{27} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{9} - \frac{4}{27} a^{8} - \frac{2}{9} a^{7} - \frac{1}{3} a^{6} + \frac{4}{9} a^{5} + \frac{4}{27} a^{4} + \frac{2}{9} a^{3} + \frac{1}{3} a^{2} - \frac{4}{9} a - \frac{4}{27}$, $\frac{1}{27} a^{21} - \frac{1}{27} a^{17} - \frac{1}{9} a^{14} + \frac{4}{27} a^{13} + \frac{1}{9} a^{12} + \frac{1}{9} a^{10} - \frac{4}{27} a^{9} - \frac{1}{9} a^{8} - \frac{1}{3} a^{7} + \frac{4}{9} a^{6} + \frac{4}{27} a^{5} + \frac{2}{9} a^{4} + \frac{1}{3} a^{3} - \frac{4}{9} a^{2} - \frac{4}{27} a - \frac{2}{9}$, $\frac{1}{1053} a^{22} + \frac{2}{117} a^{21} + \frac{4}{1053} a^{20} + \frac{4}{117} a^{19} - \frac{4}{1053} a^{18} - \frac{16}{351} a^{17} + \frac{2}{1053} a^{16} + \frac{28}{351} a^{15} + \frac{76}{1053} a^{14} + \frac{4}{39} a^{13} + \frac{133}{1053} a^{12} - \frac{19}{351} a^{11} - \frac{115}{1053} a^{10} - \frac{4}{351} a^{9} - \frac{118}{1053} a^{8} + \frac{116}{351} a^{7} - \frac{8}{81} a^{6} - \frac{5}{39} a^{5} + \frac{124}{1053} a^{4} + \frac{43}{351} a^{3} - \frac{16}{1053} a^{2} - \frac{121}{351} a + \frac{17}{1053}$, $\frac{1}{1053} a^{23} - \frac{8}{1053} a^{21} + \frac{1}{351} a^{20} + \frac{50}{1053} a^{19} + \frac{8}{351} a^{18} - \frac{31}{1053} a^{17} + \frac{1}{117} a^{16} - \frac{149}{1053} a^{15} - \frac{10}{117} a^{14} + \frac{22}{1053} a^{13} + \frac{5}{117} a^{12} - \frac{25}{1053} a^{11} - \frac{55}{351} a^{10} + \frac{137}{1053} a^{9} - \frac{8}{351} a^{8} - \frac{284}{1053} a^{7} - \frac{5}{39} a^{6} + \frac{409}{1053} a^{5} - \frac{142}{351} a^{4} + \frac{2}{1053} a^{3} - \frac{103}{351} a^{2} + \frac{38}{1053} a + \frac{41}{351}$, $\frac{1}{107417583} a^{24} + \frac{23381}{107417583} a^{23} + \frac{6718}{107417583} a^{22} - \frac{1217842}{107417583} a^{21} - \frac{367}{434889} a^{20} + \frac{3857914}{107417583} a^{19} - \frac{77491}{8262891} a^{18} + \frac{483640}{15345369} a^{17} + \frac{5611192}{107417583} a^{16} - \frac{1284019}{15345369} a^{15} - \frac{7625729}{107417583} a^{14} + \frac{6374912}{107417583} a^{13} - \frac{10424}{95823} a^{12} - \frac{142859}{15345369} a^{11} + \frac{4755575}{107417583} a^{10} + \frac{1544161}{15345369} a^{9} - \frac{7651841}{107417583} a^{8} + \frac{6823616}{107417583} a^{7} + \frac{25282735}{107417583} a^{6} - \frac{1436239}{15345369} a^{5} + \frac{1850699}{107417583} a^{4} + \frac{33925774}{107417583} a^{3} - \frac{41288398}{107417583} a^{2} + \frac{4583830}{107417583} a - \frac{14814697}{35805861}$, $\frac{1}{2448137292952961380369114895687217} a^{25} - \frac{2998958800635455157573670}{2448137292952961380369114895687217} a^{24} - \frac{31299202552457050347840247199}{2448137292952961380369114895687217} a^{23} - \frac{50696359637800002214849840280}{128849331208050598966795520825643} a^{22} + \frac{33102879175108800339750291334913}{2448137292952961380369114895687217} a^{21} - \frac{3508885327276555641159679456966}{272015254772551264485457210631913} a^{20} - \frac{62080555664724178375583267738425}{2448137292952961380369114895687217} a^{19} + \frac{2298871251859148991983126459792}{2448137292952961380369114895687217} a^{18} - \frac{115988520570106070880821315665178}{2448137292952961380369114895687217} a^{17} + \frac{1125436497030014709656062138401}{30223917196950140498384134514657} a^{16} + \frac{345516952475847548098318199854264}{2448137292952961380369114895687217} a^{15} + \frac{380511926911086445360400968516597}{2448137292952961380369114895687217} a^{14} - \frac{141155497036191901237804069810201}{2448137292952961380369114895687217} a^{13} + \frac{20658513681869727130159968662224}{272015254772551264485457210631913} a^{12} + \frac{202890556343023654139189923503917}{2448137292952961380369114895687217} a^{11} + \frac{81502309347391498995364711600793}{2448137292952961380369114895687217} a^{10} + \frac{10783970001622041996999346778380}{188318253304073952336085761206709} a^{9} + \frac{5452325452903149284257781596306}{42949777069350199655598506941881} a^{8} - \frac{16679503372428266968133275619093}{2448137292952961380369114895687217} a^{7} - \frac{1033900301606511830820105567345764}{2448137292952961380369114895687217} a^{6} + \frac{41036024070860738228332232731373}{128849331208050598966795520825643} a^{5} + \frac{64758773287011848796405199562126}{272015254772551264485457210631913} a^{4} + \frac{100312062321735173720029372498406}{349733898993280197195587842241031} a^{3} - \frac{39115732331886078717212252162824}{188318253304073952336085761206709} a^{2} - \frac{24763970074028723254994197862176}{90671751590850421495152403543971} a - \frac{1079173402391024926323521795247800}{2448137292952961380369114895687217}$

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:  \( 305647208281.4963 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{12}\cdot 305647208281.4963 \cdot 1}{2\sqrt{4863650591176530100056566961783399658203125}}\approx 1.04936763574588$ (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

\(\Q(\sqrt{5}) \), 13.1.63121332085847281.1

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

Sibling fields

Degree 26 sibling: Deg 26

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $26$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{13}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{13}$ $26$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $26$ ${\href{/LocalNumberField/29.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/31.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/41.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
631Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
1.631.2t1.a.a$1$ $ 631 $ \(\Q(\sqrt{-631}) \) $C_2$ (as 2T1) $1$ $-1$
1.3155.2t1.a.a$1$ $ 5 \cdot 631 $ \(\Q(\sqrt{-3155}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.15775.26t3.a.c$2$ $ 5^{2} \cdot 631 $ 26.2.4863650591176530100056566961783399658203125.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.631.13t2.a.c$2$ $ 631 $ 13.1.63121332085847281.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.631.13t2.a.b$2$ $ 631 $ 13.1.63121332085847281.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.15775.26t3.a.f$2$ $ 5^{2} \cdot 631 $ 26.2.4863650591176530100056566961783399658203125.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.15775.26t3.a.b$2$ $ 5^{2} \cdot 631 $ 26.2.4863650591176530100056566961783399658203125.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.631.13t2.a.a$2$ $ 631 $ 13.1.63121332085847281.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.631.13t2.a.d$2$ $ 631 $ 13.1.63121332085847281.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.631.13t2.a.f$2$ $ 631 $ 13.1.63121332085847281.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.15775.26t3.a.d$2$ $ 5^{2} \cdot 631 $ 26.2.4863650591176530100056566961783399658203125.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.15775.26t3.a.a$2$ $ 5^{2} \cdot 631 $ 26.2.4863650591176530100056566961783399658203125.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.631.13t2.a.e$2$ $ 631 $ 13.1.63121332085847281.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.15775.26t3.a.e$2$ $ 5^{2} \cdot 631 $ 26.2.4863650591176530100056566961783399658203125.1 $D_{26}$ (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 *.