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Properties

Label 26.26.7897530471...4557.1
Degree $26$
Signature $[26, 0]$
Discriminant $157^{25}$
Root discriminant $129.25$
Ramified prime $157$
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![342901, -2166125, 1141817, 21152028, -57715210, 25846692, 101902947, -148474350, -5772234, 152145203, -89817297, -45224081, 65170750, -9122158, -17012789, 7430502, 1523303, -1560168, 89472, 154495, -28151, -7628, 2185, 168, -75, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - x^25 - 75*x^24 + 168*x^23 + 2185*x^22 - 7628*x^21 - 28151*x^20 + 154495*x^19 + 89472*x^18 - 1560168*x^17 + 1523303*x^16 + 7430502*x^15 - 17012789*x^14 - 9122158*x^13 + 65170750*x^12 - 45224081*x^11 - 89817297*x^10 + 152145203*x^9 - 5772234*x^8 - 148474350*x^7 + 101902947*x^6 + 25846692*x^5 - 57715210*x^4 + 21152028*x^3 + 1141817*x^2 - 2166125*x + 342901)
 
gp: K = bnfinit(x^26 - x^25 - 75*x^24 + 168*x^23 + 2185*x^22 - 7628*x^21 - 28151*x^20 + 154495*x^19 + 89472*x^18 - 1560168*x^17 + 1523303*x^16 + 7430502*x^15 - 17012789*x^14 - 9122158*x^13 + 65170750*x^12 - 45224081*x^11 - 89817297*x^10 + 152145203*x^9 - 5772234*x^8 - 148474350*x^7 + 101902947*x^6 + 25846692*x^5 - 57715210*x^4 + 21152028*x^3 + 1141817*x^2 - 2166125*x + 342901, 1)
 

Normalized defining polynomial

\( x^{26} - x^{25} - 75 x^{24} + 168 x^{23} + 2185 x^{22} - 7628 x^{21} - 28151 x^{20} + 154495 x^{19} + 89472 x^{18} - 1560168 x^{17} + 1523303 x^{16} + 7430502 x^{15} - 17012789 x^{14} - 9122158 x^{13} + 65170750 x^{12} - 45224081 x^{11} - 89817297 x^{10} + 152145203 x^{9} - 5772234 x^{8} - 148474350 x^{7} + 101902947 x^{6} + 25846692 x^{5} - 57715210 x^{4} + 21152028 x^{3} + 1141817 x^{2} - 2166125 x + 342901 \)

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:  \(7897530471260483511971019880588399158013523966509154557=157^{25}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $129.25$
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:  $157$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(157\)
Dirichlet character group:    $\lbrace$$\chi_{157}(64,·)$, $\chi_{157}(1,·)$, $\chi_{157}(130,·)$, $\chi_{157}(67,·)$, $\chi_{157}(4,·)$, $\chi_{157}(75,·)$, $\chi_{157}(141,·)$, $\chi_{157}(14,·)$, $\chi_{157}(143,·)$, $\chi_{157}(16,·)$, $\chi_{157}(82,·)$, $\chi_{157}(153,·)$, $\chi_{157}(90,·)$, $\chi_{157}(27,·)$, $\chi_{157}(156,·)$, $\chi_{157}(93,·)$, $\chi_{157}(99,·)$, $\chi_{157}(101,·)$, $\chi_{157}(39,·)$, $\chi_{157}(108,·)$, $\chi_{157}(46,·)$, $\chi_{157}(111,·)$, $\chi_{157}(49,·)$, $\chi_{157}(118,·)$, $\chi_{157}(56,·)$, $\chi_{157}(58,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{13} a^{11} + \frac{5}{13} a^{10} + \frac{6}{13} a^{9} + \frac{3}{13} a^{7} + \frac{2}{13} a^{6} + \frac{5}{13} a^{5} - \frac{4}{13} a^{3} + \frac{6}{13} a^{2} + \frac{2}{13} a$, $\frac{1}{13} a^{12} - \frac{6}{13} a^{10} - \frac{4}{13} a^{9} + \frac{3}{13} a^{8} - \frac{5}{13} a^{6} + \frac{1}{13} a^{5} - \frac{4}{13} a^{4} - \frac{2}{13} a^{2} + \frac{3}{13} a$, $\frac{1}{13} a^{13} - \frac{1}{13} a$, $\frac{1}{13} a^{14} - \frac{1}{13} a^{2}$, $\frac{1}{13} a^{15} - \frac{1}{13} a^{3}$, $\frac{1}{13} a^{16} - \frac{1}{13} a^{4}$, $\frac{1}{13} a^{17} - \frac{1}{13} a^{5}$, $\frac{1}{13} a^{18} - \frac{1}{13} a^{6}$, $\frac{1}{13} a^{19} - \frac{1}{13} a^{7}$, $\frac{1}{13} a^{20} - \frac{1}{13} a^{8}$, $\frac{1}{13} a^{21} - \frac{1}{13} a^{9}$, $\frac{1}{169} a^{22} - \frac{3}{169} a^{21} - \frac{2}{169} a^{20} - \frac{5}{169} a^{19} + \frac{3}{169} a^{18} - \frac{5}{169} a^{17} + \frac{1}{169} a^{16} - \frac{4}{169} a^{15} - \frac{4}{169} a^{14} - \frac{3}{169} a^{13} - \frac{2}{169} a^{12} - \frac{5}{169} a^{11} - \frac{79}{169} a^{10} + \frac{46}{169} a^{9} + \frac{48}{169} a^{8} - \frac{49}{169} a^{7} - \frac{68}{169} a^{6} + \frac{30}{169} a^{5} - \frac{45}{169} a^{4} + \frac{63}{169} a^{3} - \frac{22}{169} a^{2} - \frac{5}{13} a$, $\frac{1}{52897} a^{23} - \frac{145}{52897} a^{22} - \frac{447}{52897} a^{21} + \frac{2021}{52897} a^{20} - \frac{1692}{52897} a^{19} - \frac{678}{52897} a^{18} - \frac{1031}{52897} a^{17} + \frac{1700}{52897} a^{16} - \frac{515}{52897} a^{15} - \frac{202}{52897} a^{14} - \frac{1877}{52897} a^{13} - \frac{579}{52897} a^{12} - \frac{1202}{52897} a^{11} - \frac{18272}{52897} a^{10} + \frac{17579}{52897} a^{9} - \frac{25546}{52897} a^{8} - \frac{1385}{4069} a^{7} - \frac{13610}{52897} a^{6} + \frac{5497}{52897} a^{5} + \frac{603}{52897} a^{4} - \frac{7317}{52897} a^{3} - \frac{24384}{52897} a^{2} - \frac{282}{4069} a + \frac{68}{313}$, $\frac{1}{52897} a^{24} + \frac{125}{52897} a^{22} - \frac{1446}{52897} a^{21} - \frac{50}{52897} a^{20} - \frac{1998}{52897} a^{18} + \frac{567}{52897} a^{17} - \frac{972}{52897} a^{16} + \frac{115}{4069} a^{15} + \frac{446}{52897} a^{14} + \frac{192}{52897} a^{13} + \frac{1857}{52897} a^{12} + \frac{43}{4069} a^{11} - \frac{16746}{52897} a^{10} + \frac{21600}{52897} a^{9} + \frac{16297}{52897} a^{8} - \frac{950}{4069} a^{7} + \frac{26170}{52897} a^{6} + \frac{21428}{52897} a^{5} + \frac{19709}{52897} a^{4} + \frac{1455}{4069} a^{3} + \frac{5692}{52897} a^{2} + \frac{58}{4069} a - \frac{156}{313}$, $\frac{1}{10028981764062462049899401} a^{25} - \frac{5688364163881150633}{771460135697112465376877} a^{24} - \frac{29134599427552744967}{10028981764062462049899401} a^{23} - \frac{627868385729157386578}{10028981764062462049899401} a^{22} - \frac{136623465321652403195072}{10028981764062462049899401} a^{21} - \frac{231507227365122674320492}{10028981764062462049899401} a^{20} - \frac{19307213278074480894602}{771460135697112465376877} a^{19} - \frac{230703238906491866139089}{10028981764062462049899401} a^{18} - \frac{211007056179730709123151}{10028981764062462049899401} a^{17} + \frac{268909719514306105860375}{10028981764062462049899401} a^{16} + \frac{16703750264203329103191}{10028981764062462049899401} a^{15} + \frac{4303004482587828560949}{771460135697112465376877} a^{14} + \frac{383583971622830757714272}{10028981764062462049899401} a^{13} + \frac{26674739069308264901479}{10028981764062462049899401} a^{12} + \frac{236002766130192916701536}{10028981764062462049899401} a^{11} - \frac{249529985796798245252979}{10028981764062462049899401} a^{10} + \frac{3527427075369240852748625}{10028981764062462049899401} a^{9} + \frac{1384408295327246255124369}{10028981764062462049899401} a^{8} - \frac{1642386404897163396790502}{10028981764062462049899401} a^{7} + \frac{4936649816076569004654813}{10028981764062462049899401} a^{6} - \frac{1587673967291985737443093}{10028981764062462049899401} a^{5} - \frac{1419241761395024620000657}{10028981764062462049899401} a^{4} - \frac{259124398836265688610354}{10028981764062462049899401} a^{3} + \frac{2307729151316478428984405}{10028981764062462049899401} a^{2} - \frac{44418715874596343666148}{771460135697112465376877} a + \frac{3702233239277965558724}{59343087361316343490529}$

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:  \( 68363713941558436000 \) (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

\(\Q(\sqrt{157}) \), 13.13.224282727500720205065439601.1

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 $26$ ${\href{/LocalNumberField/3.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/11.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{26}$ ${\href{/LocalNumberField/17.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/19.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/31.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/37.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/47.13.0.1}{13} }^{2}$ $26$ $26$

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
157Data not computed