Properties

Label 25.1.425...361.1
Degree $25$
Signature $[1, 12]$
Discriminant $4.258\times 10^{37}$
Root discriminant $32.00$
Ramified prime $1367$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $D_{25}$ (as 25T4)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 2*x^24 - x^23 + 13*x^22 + 50*x^21 + 3*x^20 - 29*x^19 - 5*x^18 + 128*x^17 - 146*x^16 - 239*x^15 + 32*x^14 + 747*x^13 - 262*x^12 - 1333*x^11 - 456*x^10 + 1762*x^9 + 962*x^8 - 1685*x^7 - 1308*x^6 + 1439*x^5 + 1916*x^4 - 95*x^3 - 410*x^2 + 94*x - 1)
 
gp: K = bnfinit(x^25 - 2*x^24 - x^23 + 13*x^22 + 50*x^21 + 3*x^20 - 29*x^19 - 5*x^18 + 128*x^17 - 146*x^16 - 239*x^15 + 32*x^14 + 747*x^13 - 262*x^12 - 1333*x^11 - 456*x^10 + 1762*x^9 + 962*x^8 - 1685*x^7 - 1308*x^6 + 1439*x^5 + 1916*x^4 - 95*x^3 - 410*x^2 + 94*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 94, -410, -95, 1916, 1439, -1308, -1685, 962, 1762, -456, -1333, -262, 747, 32, -239, -146, 128, -5, -29, 3, 50, 13, -1, -2, 1]);
 

\( x^{25} - 2 x^{24} - x^{23} + 13 x^{22} + 50 x^{21} + 3 x^{20} - 29 x^{19} - 5 x^{18} + 128 x^{17} - 146 x^{16} - 239 x^{15} + 32 x^{14} + 747 x^{13} - 262 x^{12} - 1333 x^{11} - 456 x^{10} + 1762 x^{9} + 962 x^{8} - 1685 x^{7} - 1308 x^{6} + 1439 x^{5} + 1916 x^{4} - 95 x^{3} - 410 x^{2} + 94 x - 1 \)

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:  \(42581619494519898305269398418425099361\)\(\medspace = 1367^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $32.00$
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:  $1367$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5} a^{18} + \frac{1}{5} a^{17} + \frac{2}{5} a^{16} + \frac{2}{5} a^{15} + \frac{1}{5} a^{13} - \frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{19} + \frac{1}{5} a^{17} - \frac{2}{5} a^{15} + \frac{1}{5} a^{14} - \frac{2}{5} a^{13} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{35} a^{20} + \frac{3}{35} a^{18} + \frac{17}{35} a^{17} - \frac{8}{35} a^{16} - \frac{3}{7} a^{15} - \frac{17}{35} a^{14} - \frac{13}{35} a^{13} - \frac{17}{35} a^{12} - \frac{16}{35} a^{11} - \frac{8}{35} a^{9} + \frac{3}{35} a^{8} + \frac{2}{5} a^{7} - \frac{8}{35} a^{6} - \frac{2}{7} a^{5} - \frac{8}{35} a^{4} + \frac{9}{35} a^{3} + \frac{1}{5} a^{2} - \frac{1}{7} a + \frac{6}{35}$, $\frac{1}{5005} a^{21} - \frac{31}{5005} a^{20} - \frac{459}{5005} a^{19} - \frac{62}{5005} a^{18} + \frac{557}{5005} a^{17} - \frac{199}{455} a^{16} - \frac{31}{143} a^{15} + \frac{122}{5005} a^{14} + \frac{519}{5005} a^{13} + \frac{296}{715} a^{12} + \frac{1707}{5005} a^{11} - \frac{694}{5005} a^{10} - \frac{1548}{5005} a^{9} + \frac{239}{1001} a^{8} + \frac{1371}{5005} a^{7} + \frac{69}{715} a^{6} - \frac{1504}{5005} a^{5} - \frac{877}{5005} a^{4} - \frac{1203}{5005} a^{3} - \frac{72}{385} a^{2} - \frac{263}{715} a + \frac{2432}{5005}$, $\frac{1}{5005} a^{22} + \frac{2}{1001} a^{20} - \frac{277}{5005} a^{19} - \frac{6}{385} a^{18} + \frac{349}{5005} a^{17} - \frac{261}{1001} a^{16} + \frac{1093}{5005} a^{15} - \frac{18}{91} a^{14} - \frac{2}{7} a^{13} - \frac{59}{715} a^{12} + \frac{2316}{5005} a^{11} - \frac{16}{77} a^{10} - \frac{5}{143} a^{9} - \frac{337}{5005} a^{8} + \frac{1943}{5005} a^{7} + \frac{2029}{5005} a^{6} + \frac{261}{5005} a^{5} - \frac{256}{715} a^{4} + \frac{1668}{5005} a^{3} + \frac{2176}{5005} a^{2} + \frac{5}{11} a - \frac{302}{715}$, $\frac{1}{95994173275} a^{23} - \frac{183483}{13713453325} a^{22} - \frac{5819479}{95994173275} a^{21} + \frac{905452656}{95994173275} a^{20} + \frac{9271386139}{95994173275} a^{19} + \frac{635793987}{19198834655} a^{18} + \frac{8474931123}{95994173275} a^{17} + \frac{47097051713}{95994173275} a^{16} - \frac{217167058}{3839766931} a^{15} + \frac{9947501383}{95994173275} a^{14} + \frac{208742447}{5646716075} a^{13} + \frac{803658604}{2742690665} a^{12} - \frac{45794007676}{95994173275} a^{11} - \frac{4130023323}{8726743025} a^{10} - \frac{28155216319}{95994173275} a^{9} + \frac{29267437406}{95994173275} a^{8} - \frac{29440808464}{95994173275} a^{7} + \frac{3835072528}{13713453325} a^{6} - \frac{16588743111}{95994173275} a^{5} + \frac{28724566809}{95994173275} a^{4} - \frac{3659491006}{19198834655} a^{3} + \frac{4492599453}{95994173275} a^{2} - \frac{22691376417}{95994173275} a - \frac{4984496639}{13713453325}$, $\frac{1}{438268897996289485300175} a^{24} + \frac{4087554299}{438268897996289485300175} a^{23} - \frac{1335679851016194917}{62609842570898497900025} a^{22} + \frac{7235192998276317241}{438268897996289485300175} a^{21} - \frac{2424775916968483872686}{438268897996289485300175} a^{20} - \frac{3588775498781502915972}{87653779599257897060035} a^{19} + \frac{1165323593434190128089}{25780523411546440311775} a^{18} - \frac{44940314213514718530197}{438268897996289485300175} a^{17} - \frac{76309377673177312}{171456978378100415} a^{16} + \frac{7426432124512626790889}{62609842570898497900025} a^{15} - \frac{7168564947558751617957}{33712992153560729638475} a^{14} + \frac{845333069389219097942}{6742598430712145927695} a^{13} + \frac{10556530561461009039542}{62609842570898497900025} a^{12} - \frac{5027421524652952696034}{25780523411546440311775} a^{11} + \frac{79301188705385465101636}{438268897996289485300175} a^{10} + \frac{7910444599381398021263}{62609842570898497900025} a^{9} - \frac{6051924722113701188833}{33712992153560729638475} a^{8} - \frac{918197819960052526666}{3682931915935205758825} a^{7} - \frac{119166946808573472036961}{438268897996289485300175} a^{6} - \frac{53503215984246121258861}{438268897996289485300175} a^{5} - \frac{30163461275676522593126}{87653779599257897060035} a^{4} - \frac{86206621467095059917757}{438268897996289485300175} a^{3} + \frac{132592247683542529946313}{438268897996289485300175} a^{2} - \frac{29876554378203335895699}{62609842570898497900025} a - \frac{6674258731291625300587}{87653779599257897060035}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 222319295.2100307 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{12}\cdot 222319295.2100307 \cdot 4}{2\sqrt{42581619494519898305269398418425099361}}\approx 0.515922515220275$ (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

5.1.1868689.1

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 $25$ $25$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $25$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{5}$ $25$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $25$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $25$ $25$

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
1367Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.1367.2t1.a.a$1$ $ 1367 $ \(\Q(\sqrt{-1367}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.1367.5t2.a.a$2$ $ 1367 $ 5.1.1868689.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.1367.5t2.a.b$2$ $ 1367 $ 5.1.1868689.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.1367.25t4.a.b$2$ $ 1367 $ 25.1.42581619494519898305269398418425099361.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.1367.25t4.a.i$2$ $ 1367 $ 25.1.42581619494519898305269398418425099361.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.1367.25t4.a.e$2$ $ 1367 $ 25.1.42581619494519898305269398418425099361.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.1367.25t4.a.g$2$ $ 1367 $ 25.1.42581619494519898305269398418425099361.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.1367.25t4.a.a$2$ $ 1367 $ 25.1.42581619494519898305269398418425099361.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.1367.25t4.a.j$2$ $ 1367 $ 25.1.42581619494519898305269398418425099361.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.1367.25t4.a.h$2$ $ 1367 $ 25.1.42581619494519898305269398418425099361.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.1367.25t4.a.f$2$ $ 1367 $ 25.1.42581619494519898305269398418425099361.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.1367.25t4.a.d$2$ $ 1367 $ 25.1.42581619494519898305269398418425099361.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.1367.25t4.a.c$2$ $ 1367 $ 25.1.42581619494519898305269398418425099361.1 $D_{25}$ (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 *.