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Properties

Label 25.5.188...376.1
Degree $25$
Signature $[5, 10]$
Discriminant $1.889\times 10^{38}$
Root discriminant $33.97$
Ramified primes $2, 41$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_5\times D_5$ (as 25T3)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 2*x^24 - 2*x^23 - 10*x^22 + 33*x^21 + 5*x^20 - 6*x^19 - 115*x^18 - 16*x^17 + 168*x^16 + 250*x^15 - 16*x^14 + 291*x^13 - 1042*x^12 + 750*x^11 - 1666*x^10 + 1541*x^9 - 903*x^8 + 418*x^7 + 161*x^6 - 27*x^5 + 92*x^4 + 3*x^3 + 8*x^2 + 4*x - 1)
 
gp: K = bnfinit(x^25 - 2*x^24 - 2*x^23 - 10*x^22 + 33*x^21 + 5*x^20 - 6*x^19 - 115*x^18 - 16*x^17 + 168*x^16 + 250*x^15 - 16*x^14 + 291*x^13 - 1042*x^12 + 750*x^11 - 1666*x^10 + 1541*x^9 - 903*x^8 + 418*x^7 + 161*x^6 - 27*x^5 + 92*x^4 + 3*x^3 + 8*x^2 + 4*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 4, 8, 3, 92, -27, 161, 418, -903, 1541, -1666, 750, -1042, 291, -16, 250, 168, -16, -115, -6, 5, 33, -10, -2, -2, 1]);
 

\( x^{25} - 2 x^{24} - 2 x^{23} - 10 x^{22} + 33 x^{21} + 5 x^{20} - 6 x^{19} - 115 x^{18} - 16 x^{17} + 168 x^{16} + 250 x^{15} - 16 x^{14} + 291 x^{13} - 1042 x^{12} + 750 x^{11} - 1666 x^{10} + 1541 x^{9} - 903 x^{8} + 418 x^{7} + 161 x^{6} - 27 x^{5} + 92 x^{4} + 3 x^{3} + 8 x^{2} + 4 x - 1 \)

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

Invariants

Degree:  $25$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[5, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(188919613181312032574569023867244773376\)\(\medspace = 2^{20}\cdot 41^{20}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $33.97$
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:  $2, 41$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $5$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{12} + \frac{1}{9} a^{10} - \frac{1}{9} a^{8} - \frac{4}{9} a^{6} + \frac{4}{9} a^{4} - \frac{4}{9} a^{2} + \frac{4}{9}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{13} + \frac{1}{9} a^{11} - \frac{1}{9} a^{9} - \frac{4}{9} a^{7} + \frac{4}{9} a^{5} - \frac{4}{9} a^{3} + \frac{4}{9} a$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{8} - \frac{2}{9}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{9} - \frac{2}{9} a$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{10} - \frac{2}{9} a^{2}$, $\frac{1}{27} a^{19} + \frac{1}{27} a^{17} - \frac{1}{27} a^{16} + \frac{1}{9} a^{13} + \frac{1}{27} a^{11} - \frac{1}{9} a^{10} - \frac{2}{27} a^{9} + \frac{2}{27} a^{8} - \frac{1}{3} a^{6} + \frac{2}{9} a^{5} + \frac{1}{3} a^{4} - \frac{2}{27} a^{3} + \frac{4}{9} a^{2} - \frac{8}{27} a - \frac{10}{27}$, $\frac{1}{27} a^{20} + \frac{1}{27} a^{18} - \frac{1}{27} a^{17} + \frac{4}{27} a^{12} - \frac{1}{9} a^{11} + \frac{4}{27} a^{10} + \frac{2}{27} a^{9} + \frac{1}{9} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{13}{27} a^{4} + \frac{4}{9} a^{3} - \frac{5}{27} a^{2} - \frac{10}{27} a - \frac{4}{9}$, $\frac{1}{27} a^{21} - \frac{1}{27} a^{18} - \frac{1}{27} a^{17} + \frac{1}{27} a^{16} + \frac{1}{27} a^{13} - \frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{4}{27} a^{10} - \frac{4}{27} a^{9} - \frac{2}{27} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{7}{27} a^{5} + \frac{1}{9} a^{4} - \frac{1}{9} a^{3} - \frac{13}{27} a^{2} + \frac{5}{27} a + \frac{1}{27}$, $\frac{1}{27} a^{22} - \frac{1}{27} a^{18} - \frac{1}{27} a^{17} - \frac{1}{27} a^{16} + \frac{1}{27} a^{14} + \frac{1}{9} a^{12} - \frac{1}{9} a^{11} + \frac{2}{27} a^{10} + \frac{2}{27} a^{9} + \frac{2}{27} a^{8} - \frac{1}{3} a^{7} - \frac{2}{27} a^{6} + \frac{1}{3} a^{5} + \frac{2}{9} a^{4} + \frac{4}{9} a^{3} + \frac{8}{27} a^{2} - \frac{10}{27} a + \frac{8}{27}$, $\frac{1}{243} a^{23} - \frac{1}{243} a^{22} - \frac{1}{243} a^{21} - \frac{4}{243} a^{20} - \frac{1}{81} a^{18} + \frac{1}{27} a^{17} - \frac{13}{243} a^{16} - \frac{2}{243} a^{15} + \frac{5}{243} a^{14} + \frac{8}{243} a^{13} + \frac{11}{243} a^{12} + \frac{7}{81} a^{11} - \frac{1}{9} a^{9} - \frac{1}{243} a^{8} + \frac{91}{243} a^{7} - \frac{58}{243} a^{6} - \frac{25}{243} a^{5} + \frac{83}{243} a^{4} + \frac{29}{81} a^{3} + \frac{25}{81} a^{2} - \frac{1}{3} a - \frac{13}{243}$, $\frac{1}{63037299496134907413} a^{24} + \frac{9427940055707662}{21012433165378302471} a^{23} + \frac{358581875759631427}{63037299496134907413} a^{22} + \frac{45485323642375102}{63037299496134907413} a^{21} + \frac{844871820261555629}{63037299496134907413} a^{20} + \frac{250166720875192742}{21012433165378302471} a^{19} - \frac{38592183794982238}{21012433165378302471} a^{18} - \frac{1739799531437674297}{63037299496134907413} a^{17} - \frac{281224838037817463}{7004144388459434157} a^{16} + \frac{1078413241808969117}{21012433165378302471} a^{15} - \frac{71256602451808061}{63037299496134907413} a^{14} + \frac{1191123346874256475}{63037299496134907413} a^{13} - \frac{332133245184295717}{63037299496134907413} a^{12} - \frac{1585551273144235412}{21012433165378302471} a^{11} + \frac{30045630546637568}{2334714796153144719} a^{10} - \frac{6567243915847655107}{63037299496134907413} a^{9} - \frac{3208842231129570820}{21012433165378302471} a^{8} - \frac{1601735155552745030}{7004144388459434157} a^{7} + \frac{11801431537411172800}{63037299496134907413} a^{6} + \frac{13576898936438163235}{63037299496134907413} a^{5} - \frac{1884015453554317759}{63037299496134907413} a^{4} + \frac{770768142286787159}{2334714796153144719} a^{3} + \frac{7869917667140260525}{21012433165378302471} a^{2} + \frac{19910342011098244208}{63037299496134907413} a + \frac{12783195893832686228}{63037299496134907413}$

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:  $14$
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:  \( 28022710779.842506 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{5}\cdot(2\pi)^{10}\cdot 28022710779.842506 \cdot 1}{2\sqrt{188919613181312032574569023867244773376}}\approx 3.12816956924097$ (assuming GRH)

Galois group

$C_5\times D_5$ (as 25T3):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 50
The 20 conjugacy class representatives for $C_5\times D_5$
Character table for $C_5\times D_5$

Intermediate fields

5.5.2825761.1, 5.1.45212176.1

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

Sibling fields

Degree 10 siblings: data not computed

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{5}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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
$2$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
41Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.4.2t1.a.a$1$ $ 2^{2}$ \(\Q(\sqrt{-1}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.41.5t1.a.b$1$ $ 41 $ 5.5.2825761.1 $C_5$ (as 5T1) $0$ $1$
* 1.41.5t1.a.c$1$ $ 41 $ 5.5.2825761.1 $C_5$ (as 5T1) $0$ $1$
* 1.41.5t1.a.a$1$ $ 41 $ 5.5.2825761.1 $C_5$ (as 5T1) $0$ $1$
1.164.10t1.a.d$1$ $ 2^{2} \cdot 41 $ 10.0.8176563434619904.1 $C_{10}$ (as 10T1) $0$ $-1$
1.164.10t1.a.a$1$ $ 2^{2} \cdot 41 $ 10.0.8176563434619904.1 $C_{10}$ (as 10T1) $0$ $-1$
* 1.41.5t1.a.d$1$ $ 41 $ 5.5.2825761.1 $C_5$ (as 5T1) $0$ $1$
1.164.10t1.a.c$1$ $ 2^{2} \cdot 41 $ 10.0.8176563434619904.1 $C_{10}$ (as 10T1) $0$ $-1$
1.164.10t1.a.b$1$ $ 2^{2} \cdot 41 $ 10.0.8176563434619904.1 $C_{10}$ (as 10T1) $0$ $-1$
* 2.6724.10t6.a.d$2$ $ 2^{2} \cdot 41^{2}$ 25.5.188919613181312032574569023867244773376.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.6724.10t6.a.c$2$ $ 2^{2} \cdot 41^{2}$ 25.5.188919613181312032574569023867244773376.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.164.10t6.a.b$2$ $ 2^{2} \cdot 41 $ 25.5.188919613181312032574569023867244773376.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.164.10t6.a.a$2$ $ 2^{2} \cdot 41 $ 25.5.188919613181312032574569023867244773376.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.6724.5t2.a.b$2$ $ 2^{2} \cdot 41^{2}$ 5.1.45212176.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.6724.10t6.a.b$2$ $ 2^{2} \cdot 41^{2}$ 25.5.188919613181312032574569023867244773376.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.164.10t6.a.d$2$ $ 2^{2} \cdot 41 $ 25.5.188919613181312032574569023867244773376.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.6724.5t2.a.a$2$ $ 2^{2} \cdot 41^{2}$ 5.1.45212176.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.164.10t6.a.c$2$ $ 2^{2} \cdot 41 $ 25.5.188919613181312032574569023867244773376.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.6724.10t6.a.a$2$ $ 2^{2} \cdot 41^{2}$ 25.5.188919613181312032574569023867244773376.1 $C_5\times D_5$ (as 25T3) $0$ $0$

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 *.