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Properties

Label 26.26.1326754903...9328.1
Degree $26$
Signature $[26, 0]$
Discriminant $2^{39}\cdot 53^{24}$
Root discriminant $110.45$
Ramified primes $2, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{26}$ (as 26T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-12167, -42320, 501952, 1587526, -4454532, -8893984, 16012693, 20368692, -28852591, -24941144, 29677420, 18209442, -18730676, -8328084, 7531107, 2432394, -1957340, -453164, 327544, 52714, -34469, -3640, 2166, 134, -73, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 2*x^25 - 73*x^24 + 134*x^23 + 2166*x^22 - 3640*x^21 - 34469*x^20 + 52714*x^19 + 327544*x^18 - 453164*x^17 - 1957340*x^16 + 2432394*x^15 + 7531107*x^14 - 8328084*x^13 - 18730676*x^12 + 18209442*x^11 + 29677420*x^10 - 24941144*x^9 - 28852591*x^8 + 20368692*x^7 + 16012693*x^6 - 8893984*x^5 - 4454532*x^4 + 1587526*x^3 + 501952*x^2 - 42320*x - 12167)
 
gp: K = bnfinit(x^26 - 2*x^25 - 73*x^24 + 134*x^23 + 2166*x^22 - 3640*x^21 - 34469*x^20 + 52714*x^19 + 327544*x^18 - 453164*x^17 - 1957340*x^16 + 2432394*x^15 + 7531107*x^14 - 8328084*x^13 - 18730676*x^12 + 18209442*x^11 + 29677420*x^10 - 24941144*x^9 - 28852591*x^8 + 20368692*x^7 + 16012693*x^6 - 8893984*x^5 - 4454532*x^4 + 1587526*x^3 + 501952*x^2 - 42320*x - 12167, 1)
 

Normalized defining polynomial

\( x^{26} - 2 x^{25} - 73 x^{24} + 134 x^{23} + 2166 x^{22} - 3640 x^{21} - 34469 x^{20} + 52714 x^{19} + 327544 x^{18} - 453164 x^{17} - 1957340 x^{16} + 2432394 x^{15} + 7531107 x^{14} - 8328084 x^{13} - 18730676 x^{12} + 18209442 x^{11} + 29677420 x^{10} - 24941144 x^{9} - 28852591 x^{8} + 20368692 x^{7} + 16012693 x^{6} - 8893984 x^{5} - 4454532 x^{4} + 1587526 x^{3} + 501952 x^{2} - 42320 x - 12167 \)

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

Invariants

Degree:  $26$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[26, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(132675490325051365459874809737094631268103216670179328=2^{39}\cdot 53^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $110.45$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(424=2^{3}\cdot 53\)
Dirichlet character group:    $\lbrace$$\chi_{424}(1,·)$, $\chi_{424}(261,·)$, $\chi_{424}(225,·)$, $\chi_{424}(201,·)$, $\chi_{424}(417,·)$, $\chi_{424}(13,·)$, $\chi_{424}(77,·)$, $\chi_{424}(333,·)$, $\chi_{424}(81,·)$, $\chi_{424}(205,·)$, $\chi_{424}(213,·)$, $\chi_{424}(281,·)$, $\chi_{424}(153,·)$, $\chi_{424}(413,·)$, $\chi_{424}(69,·)$, $\chi_{424}(289,·)$, $\chi_{424}(293,·)$, $\chi_{424}(97,·)$, $\chi_{424}(169,·)$, $\chi_{424}(301,·)$, $\chi_{424}(49,·)$, $\chi_{424}(365,·)$, $\chi_{424}(309,·)$, $\chi_{424}(89,·)$, $\chi_{424}(121,·)$, $\chi_{424}(381,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{23} a^{18} - \frac{3}{23} a^{17} + \frac{11}{23} a^{16} - \frac{5}{23} a^{15} + \frac{8}{23} a^{14} - \frac{7}{23} a^{13} - \frac{4}{23} a^{12} - \frac{8}{23} a^{11} - \frac{10}{23} a^{10} - \frac{1}{23} a^{9} - \frac{7}{23} a^{8} - \frac{2}{23} a^{7} - \frac{1}{23} a^{6} + \frac{2}{23} a^{5} + \frac{1}{23} a^{4} + \frac{4}{23} a^{3} + \frac{11}{23} a^{2} + \frac{7}{23} a$, $\frac{1}{23} a^{19} + \frac{2}{23} a^{17} + \frac{5}{23} a^{16} - \frac{7}{23} a^{15} - \frac{6}{23} a^{14} - \frac{2}{23} a^{13} + \frac{3}{23} a^{12} - \frac{11}{23} a^{11} - \frac{8}{23} a^{10} - \frac{10}{23} a^{9} - \frac{7}{23} a^{7} - \frac{1}{23} a^{6} + \frac{7}{23} a^{5} + \frac{7}{23} a^{4} - \frac{6}{23} a^{2} - \frac{2}{23} a$, $\frac{1}{23} a^{20} + \frac{11}{23} a^{17} - \frac{6}{23} a^{16} + \frac{4}{23} a^{15} + \frac{5}{23} a^{14} - \frac{6}{23} a^{13} - \frac{3}{23} a^{12} + \frac{8}{23} a^{11} + \frac{10}{23} a^{10} + \frac{2}{23} a^{9} + \frac{7}{23} a^{8} + \frac{3}{23} a^{7} + \frac{9}{23} a^{6} + \frac{3}{23} a^{5} - \frac{2}{23} a^{4} + \frac{9}{23} a^{3} - \frac{1}{23} a^{2} + \frac{9}{23} a$, $\frac{1}{23} a^{21} + \frac{4}{23} a^{17} - \frac{2}{23} a^{16} - \frac{9}{23} a^{15} - \frac{2}{23} a^{14} + \frac{5}{23} a^{13} + \frac{6}{23} a^{12} + \frac{6}{23} a^{11} - \frac{3}{23} a^{10} - \frac{5}{23} a^{9} + \frac{11}{23} a^{8} + \frac{8}{23} a^{7} - \frac{9}{23} a^{6} - \frac{1}{23} a^{5} - \frac{2}{23} a^{4} + \frac{1}{23} a^{3} + \frac{3}{23} a^{2} - \frac{8}{23} a$, $\frac{1}{23} a^{22} + \frac{10}{23} a^{17} - \frac{7}{23} a^{16} - \frac{5}{23} a^{15} - \frac{4}{23} a^{14} + \frac{11}{23} a^{13} - \frac{1}{23} a^{12} + \frac{6}{23} a^{11} - \frac{11}{23} a^{10} - \frac{8}{23} a^{9} - \frac{10}{23} a^{8} - \frac{1}{23} a^{7} + \frac{3}{23} a^{6} - \frac{10}{23} a^{5} - \frac{3}{23} a^{4} + \frac{10}{23} a^{3} - \frac{6}{23} a^{2} - \frac{5}{23} a$, $\frac{1}{23} a^{23} - \frac{1}{23} a$, $\frac{1}{10007415161} a^{24} + \frac{176153190}{10007415161} a^{23} + \frac{167546221}{10007415161} a^{22} + \frac{125519760}{10007415161} a^{21} - \frac{105610713}{10007415161} a^{20} - \frac{212508789}{10007415161} a^{19} + \frac{15085601}{10007415161} a^{18} + \frac{788797394}{10007415161} a^{17} - \frac{4713665845}{10007415161} a^{16} + \frac{1814832448}{10007415161} a^{15} + \frac{2674096457}{10007415161} a^{14} - \frac{4555053797}{10007415161} a^{13} - \frac{4785869927}{10007415161} a^{12} - \frac{2232725445}{10007415161} a^{11} - \frac{3769709055}{10007415161} a^{10} + \frac{2512728723}{10007415161} a^{9} - \frac{3279458065}{10007415161} a^{8} + \frac{403753381}{10007415161} a^{7} + \frac{2707440832}{10007415161} a^{6} + \frac{2458506826}{10007415161} a^{5} + \frac{3939347686}{10007415161} a^{4} - \frac{2761605158}{10007415161} a^{3} - \frac{94462382}{10007415161} a^{2} + \frac{38434236}{435105007} a + \frac{3146874}{18917609}$, $\frac{1}{93554463095389318627211512018619778264634539822866471839} a^{25} + \frac{1840571246284937735886448388899337834706146536}{93554463095389318627211512018619778264634539822866471839} a^{24} - \frac{630921778774404753810104660171778837956190189707386259}{93554463095389318627211512018619778264634539822866471839} a^{23} + \frac{479690138734312100134196294741588124778376379040042981}{93554463095389318627211512018619778264634539822866471839} a^{22} + \frac{1197060436016804018235612034440817295201074630882064514}{93554463095389318627211512018619778264634539822866471839} a^{21} - \frac{1896311183482953895967811351306771131978867989598900916}{93554463095389318627211512018619778264634539822866471839} a^{20} + \frac{50134406488300807071862835811081420445114904609278591}{4067585351973448635965717913853033837592806079255063993} a^{19} + \frac{750266210075804271097210319430582381222057385045526451}{93554463095389318627211512018619778264634539822866471839} a^{18} + \frac{5711009241933257357071046963051411277258296123328731234}{93554463095389318627211512018619778264634539822866471839} a^{17} + \frac{7671487442847444216393822481386915095134921279245597356}{93554463095389318627211512018619778264634539822866471839} a^{16} - \frac{12428466930730400640424587417644659446184739437064550472}{93554463095389318627211512018619778264634539822866471839} a^{15} + \frac{8879985190047834394176868835176079893957189366509398920}{93554463095389318627211512018619778264634539822866471839} a^{14} + \frac{25153089977636867375801238664735529290452069050771709602}{93554463095389318627211512018619778264634539822866471839} a^{13} + \frac{21002716000143312305751123917592437477216633708853072301}{93554463095389318627211512018619778264634539822866471839} a^{12} - \frac{27475057361937321010891243485394242677340439801325649188}{93554463095389318627211512018619778264634539822866471839} a^{11} - \frac{24262415472784879505350144687315415611377697718653301902}{93554463095389318627211512018619778264634539822866471839} a^{10} - \frac{38513967944266528906994877578017687527891192999692902975}{93554463095389318627211512018619778264634539822866471839} a^{9} - \frac{2281564621664862002581118024139405653653498445833102803}{93554463095389318627211512018619778264634539822866471839} a^{8} + \frac{33191619743790646125492408910906633887459133805740109023}{93554463095389318627211512018619778264634539822866471839} a^{7} + \frac{30451601070830642938516626065031904419502865816075590051}{93554463095389318627211512018619778264634539822866471839} a^{6} - \frac{5623846020450984184129212761092728435129421177721466544}{93554463095389318627211512018619778264634539822866471839} a^{5} - \frac{29627850797894863460457645772946797461901705701511123377}{93554463095389318627211512018619778264634539822866471839} a^{4} - \frac{12033180664003220717964377691997461020725865992772763745}{93554463095389318627211512018619778264634539822866471839} a^{3} + \frac{13019329849000949854859119677942535138096939829312285658}{93554463095389318627211512018619778264634539822866471839} a^{2} - \frac{1679830851565274801710194616192642217718047522463048059}{4067585351973448635965717913853033837592806079255063993} a + \frac{43773582418237291961790744689340944586221305285844346}{176851537042323853737639909297957992938817655619785391}$

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

Class group and class number

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

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $25$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3196294146983080400 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{26}$ (as 26T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 26
The 26 conjugacy class representatives for $C_{26}$
Character table for $C_{26}$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 13.13.491258904256726154641.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $26$ $26$ ${\href{/LocalNumberField/7.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/17.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{26}$ $26$ ${\href{/LocalNumberField/31.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/41.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/47.13.0.1}{13} }^{2}$ R $26$

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

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
53Data not computed