Properties

Label 25.1.121...625.1
Degree $25$
Signature $[1, 12]$
Discriminant $1.219\times 10^{42}$
Root discriminant $48.24$
Ramified primes $5, 643$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{25}$ (as 25T4)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 11*x^24 + 42*x^23 - 37*x^22 - 156*x^21 + 345*x^20 + 267*x^19 - 2248*x^18 + 5531*x^17 - 338*x^16 - 41434*x^15 + 71882*x^14 + 112392*x^13 - 407865*x^12 + 80824*x^11 + 717928*x^10 - 228258*x^9 - 1185331*x^8 + 280801*x^7 + 1733675*x^6 - 74160*x^5 - 2202750*x^4 - 202350*x^3 + 1981500*x^2 + 536000*x - 1401875)
 
gp: K = bnfinit(x^25 - 11*x^24 + 42*x^23 - 37*x^22 - 156*x^21 + 345*x^20 + 267*x^19 - 2248*x^18 + 5531*x^17 - 338*x^16 - 41434*x^15 + 71882*x^14 + 112392*x^13 - 407865*x^12 + 80824*x^11 + 717928*x^10 - 228258*x^9 - 1185331*x^8 + 280801*x^7 + 1733675*x^6 - 74160*x^5 - 2202750*x^4 - 202350*x^3 + 1981500*x^2 + 536000*x - 1401875, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1401875, 536000, 1981500, -202350, -2202750, -74160, 1733675, 280801, -1185331, -228258, 717928, 80824, -407865, 112392, 71882, -41434, -338, 5531, -2248, 267, 345, -156, -37, 42, -11, 1]);
 

\( x^{25} - 11 x^{24} + 42 x^{23} - 37 x^{22} - 156 x^{21} + 345 x^{20} + 267 x^{19} - 2248 x^{18} + 5531 x^{17} - 338 x^{16} - 41434 x^{15} + 71882 x^{14} + 112392 x^{13} - 407865 x^{12} + 80824 x^{11} + 717928 x^{10} - 228258 x^{9} - 1185331 x^{8} + 280801 x^{7} + 1733675 x^{6} - 74160 x^{5} - 2202750 x^{4} - 202350 x^{3} + 1981500 x^{2} + 536000 x - 1401875 \)

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

Invariants

Degree:  $25$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[1, 12]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(1219471702607488515548387495680371337890625\)\(\medspace = 5^{12}\cdot 643^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $48.24$
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:  $5, 643$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{7} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{8} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{6} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{15} + \frac{2}{5} a^{7} + \frac{2}{5} a^{3}$, $\frac{1}{25} a^{16} - \frac{2}{25} a^{15} + \frac{2}{25} a^{14} - \frac{2}{25} a^{13} - \frac{1}{25} a^{12} - \frac{1}{25} a^{11} + \frac{1}{25} a^{10} - \frac{1}{25} a^{9} - \frac{11}{25} a^{8} + \frac{3}{25} a^{7} + \frac{12}{25} a^{6} + \frac{3}{25} a^{5} + \frac{11}{25} a^{4} + \frac{2}{5} a^{2}$, $\frac{1}{25} a^{17} - \frac{2}{25} a^{15} + \frac{2}{25} a^{14} + \frac{2}{25} a^{12} - \frac{1}{25} a^{11} + \frac{1}{25} a^{10} + \frac{2}{25} a^{9} - \frac{4}{25} a^{8} - \frac{7}{25} a^{7} + \frac{2}{25} a^{6} - \frac{3}{25} a^{5} + \frac{2}{25} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{25} a^{18} - \frac{2}{25} a^{15} - \frac{1}{25} a^{14} - \frac{2}{25} a^{13} + \frac{2}{25} a^{12} - \frac{1}{25} a^{11} - \frac{1}{25} a^{10} - \frac{1}{25} a^{9} + \frac{6}{25} a^{8} - \frac{12}{25} a^{7} - \frac{9}{25} a^{6} + \frac{3}{25} a^{5} - \frac{8}{25} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{175} a^{19} - \frac{1}{175} a^{18} - \frac{3}{175} a^{17} + \frac{2}{175} a^{16} - \frac{11}{175} a^{15} + \frac{6}{175} a^{14} - \frac{4}{175} a^{13} + \frac{12}{175} a^{12} + \frac{4}{175} a^{11} - \frac{4}{175} a^{10} - \frac{13}{175} a^{9} - \frac{8}{35} a^{8} - \frac{54}{175} a^{7} + \frac{9}{175} a^{6} - \frac{11}{35} a^{5} - \frac{74}{175} a^{4} + \frac{1}{5} a^{3} - \frac{17}{35} a^{2} - \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{875} a^{20} + \frac{2}{875} a^{19} + \frac{3}{175} a^{18} - \frac{1}{125} a^{17} - \frac{1}{175} a^{16} - \frac{34}{875} a^{15} - \frac{1}{125} a^{14} - \frac{1}{125} a^{13} - \frac{23}{875} a^{12} - \frac{13}{875} a^{11} - \frac{81}{875} a^{10} - \frac{13}{175} a^{9} + \frac{57}{875} a^{8} - \frac{11}{175} a^{7} - \frac{11}{125} a^{6} + \frac{279}{875} a^{5} - \frac{1}{175} a^{4} + \frac{12}{35} a^{3} - \frac{3}{7} a^{2} + \frac{9}{35} a + \frac{2}{7}$, $\frac{1}{875} a^{21} + \frac{1}{875} a^{19} + \frac{8}{875} a^{18} + \frac{4}{875} a^{17} - \frac{9}{875} a^{16} - \frac{74}{875} a^{15} + \frac{87}{875} a^{14} + \frac{66}{875} a^{13} + \frac{53}{875} a^{12} + \frac{9}{175} a^{11} - \frac{73}{875} a^{10} + \frac{2}{875} a^{9} - \frac{47}{125} a^{8} - \frac{22}{875} a^{7} + \frac{4}{125} a^{6} + \frac{302}{875} a^{5} - \frac{9}{25} a^{4} + \frac{2}{7} a^{3} + \frac{17}{35} a^{2} + \frac{12}{35} a + \frac{1}{7}$, $\frac{1}{875} a^{22} + \frac{1}{875} a^{19} - \frac{6}{875} a^{18} + \frac{13}{875} a^{17} - \frac{9}{875} a^{16} + \frac{36}{875} a^{15} + \frac{8}{875} a^{14} - \frac{12}{175} a^{13} - \frac{62}{875} a^{12} + \frac{1}{35} a^{11} - \frac{2}{875} a^{10} + \frac{81}{875} a^{9} - \frac{124}{875} a^{8} - \frac{312}{875} a^{7} - \frac{51}{875} a^{6} + \frac{416}{875} a^{5} - \frac{1}{175} a^{4} - \frac{9}{35} a^{3} - \frac{1}{7} a^{2} + \frac{6}{35} a - \frac{3}{7}$, $\frac{1}{914375} a^{23} - \frac{27}{130625} a^{22} + \frac{74}{130625} a^{21} + \frac{183}{914375} a^{20} + \frac{2612}{914375} a^{19} - \frac{8989}{914375} a^{18} + \frac{4457}{914375} a^{17} - \frac{1726}{182875} a^{16} + \frac{18719}{914375} a^{15} - \frac{2637}{182875} a^{14} - \frac{58148}{914375} a^{13} - \frac{45664}{914375} a^{12} + \frac{11772}{914375} a^{11} + \frac{4568}{914375} a^{10} - \frac{88947}{914375} a^{9} + \frac{12294}{48125} a^{8} + \frac{432346}{914375} a^{7} - \frac{72406}{182875} a^{6} + \frac{84979}{182875} a^{5} + \frac{129}{3325} a^{4} - \frac{3834}{36575} a^{3} + \frac{2594}{7315} a^{2} + \frac{2034}{7315} a - \frac{58}{209}$, $\frac{1}{7032120241317874349862747172221045400116768125} a^{24} + \frac{163920236424233816848951656575002233342}{1406424048263574869972549434444209080023353625} a^{23} - \frac{10175833450478359938963028807551168480518}{7032120241317874349862747172221045400116768125} a^{22} + \frac{631302742416154831591125425846887254313513}{1406424048263574869972549434444209080023353625} a^{21} + \frac{388297337322688714742227427847842930285897}{1004588605902553478551821024603006485730966875} a^{20} + \frac{2681906220677219058507788797559941994486849}{7032120241317874349862747172221045400116768125} a^{19} - \frac{4202384270082431656990724846795444300517589}{639283658301624940896613379292822309101524375} a^{18} + \frac{88420552988514400485277788556475814560224213}{7032120241317874349862747172221045400116768125} a^{17} - \frac{105818685483136171269620264009804922875056126}{7032120241317874349862747172221045400116768125} a^{16} - \frac{50082466498813264613094465904324382163378004}{7032120241317874349862747172221045400116768125} a^{15} + \frac{40938651271554257613489642299609761924698012}{7032120241317874349862747172221045400116768125} a^{14} + \frac{389787375383225005926254283412662978076639209}{7032120241317874349862747172221045400116768125} a^{13} + \frac{669956397186365183896231880055066231360567986}{7032120241317874349862747172221045400116768125} a^{12} - \frac{608950091897389053142369336549973591076552329}{7032120241317874349862747172221045400116768125} a^{11} - \frac{66698020083121918853221552901935186954046458}{1406424048263574869972549434444209080023353625} a^{10} + \frac{1092032586402293236918995095791484829811579}{91326236900232134413801911327546044157360625} a^{9} + \frac{672265933666085945674572187442473764209886487}{1406424048263574869972549434444209080023353625} a^{8} - \frac{2986862251068473747522996306051470190364887251}{7032120241317874349862747172221045400116768125} a^{7} + \frac{798791402248430676434351765750272836609074}{5114269266412999527172907034342578472812195} a^{6} - \frac{256108205557872183209707133460942173531007704}{1406424048263574869972549434444209080023353625} a^{5} + \frac{16472615943138208351548314262041635275420634}{40183544236102139142072840984120259429238675} a^{4} + \frac{3474021212389261540482021588397402842106822}{40183544236102139142072840984120259429238675} a^{3} - \frac{21308681566890012292728764647820097245071677}{56256961930542994798901977377768363200934145} a^{2} - \frac{26827877004292823634166721008503685980242049}{56256961930542994798901977377768363200934145} a + \frac{133586844875132207462325452355799621054403}{11251392386108598959780395475553672640186829}$

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:  $12$
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:  \( 1237128997505.3247 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{12}\cdot 1237128997505.3247 \cdot 1}{2\sqrt{1219471702607488515548387495680371337890625}}\approx 4.24118818936391$ (assuming GRH)

Galois group

$D_{25}$ (as 25T4):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 50
The 14 conjugacy class representatives for $D_{25}$
Character table for $D_{25}$

Intermediate fields

5.1.10336225.1

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 $25$ $25$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $25$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $25$ $25$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $25$ $25$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
643Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.3215.2t1.a.a$1$ $ 5 \cdot 643 $ \(\Q(\sqrt{-3215}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.3215.5t2.a.b$2$ $ 5 \cdot 643 $ 5.1.10336225.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.3215.5t2.a.a$2$ $ 5 \cdot 643 $ 5.1.10336225.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.3215.25t4.a.g$2$ $ 5 \cdot 643 $ 25.1.1219471702607488515548387495680371337890625.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.3215.25t4.a.h$2$ $ 5 \cdot 643 $ 25.1.1219471702607488515548387495680371337890625.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.3215.25t4.a.i$2$ $ 5 \cdot 643 $ 25.1.1219471702607488515548387495680371337890625.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.3215.25t4.a.f$2$ $ 5 \cdot 643 $ 25.1.1219471702607488515548387495680371337890625.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.3215.25t4.a.e$2$ $ 5 \cdot 643 $ 25.1.1219471702607488515548387495680371337890625.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.3215.25t4.a.b$2$ $ 5 \cdot 643 $ 25.1.1219471702607488515548387495680371337890625.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.3215.25t4.a.d$2$ $ 5 \cdot 643 $ 25.1.1219471702607488515548387495680371337890625.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.3215.25t4.a.c$2$ $ 5 \cdot 643 $ 25.1.1219471702607488515548387495680371337890625.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.3215.25t4.a.a$2$ $ 5 \cdot 643 $ 25.1.1219471702607488515548387495680371337890625.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.3215.25t4.a.j$2$ $ 5 \cdot 643 $ 25.1.1219471702607488515548387495680371337890625.1 $D_{25}$ (as 25T4) $1$ $0$

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 *.