Properties

Label 25.5.300...849.1
Degree $25$
Signature $[5, 10]$
Discriminant $3.005\times 10^{40}$
Root discriminant $41.60$
Ramified primes $3, 61$
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 - 3*x^24 + x^23 - 15*x^22 + 13*x^21 - 5*x^20 + 130*x^19 - 132*x^18 - 21*x^17 + 437*x^16 - 1713*x^15 + 747*x^14 + 4817*x^13 - 3107*x^12 - 7887*x^11 + 4826*x^10 + 5380*x^9 - 3452*x^8 + 864*x^7 - 1052*x^6 + 266*x^5 - 67*x^4 + 75*x^3 - 10*x^2 + 9*x - 1)
 
gp: K = bnfinit(x^25 - 3*x^24 + x^23 - 15*x^22 + 13*x^21 - 5*x^20 + 130*x^19 - 132*x^18 - 21*x^17 + 437*x^16 - 1713*x^15 + 747*x^14 + 4817*x^13 - 3107*x^12 - 7887*x^11 + 4826*x^10 + 5380*x^9 - 3452*x^8 + 864*x^7 - 1052*x^6 + 266*x^5 - 67*x^4 + 75*x^3 - 10*x^2 + 9*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 9, -10, 75, -67, 266, -1052, 864, -3452, 5380, 4826, -7887, -3107, 4817, 747, -1713, 437, -21, -132, 130, -5, 13, -15, 1, -3, 1]);
 

\( x^{25} - 3 x^{24} + x^{23} - 15 x^{22} + 13 x^{21} - 5 x^{20} + 130 x^{19} - 132 x^{18} - 21 x^{17} + 437 x^{16} - 1713 x^{15} + 747 x^{14} + 4817 x^{13} - 3107 x^{12} - 7887 x^{11} + 4826 x^{10} + 5380 x^{9} - 3452 x^{8} + 864 x^{7} - 1052 x^{6} + 266 x^{5} - 67 x^{4} + 75 x^{3} - 10 x^{2} + 9 x - 1 \)

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:  \(30047562514932432943626078739673704593849\)\(\medspace = 3^{10}\cdot 61^{20}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $41.60$
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:  $3, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{7} a^{20} - \frac{2}{7} a^{19} + \frac{2}{7} a^{18} + \frac{1}{7} a^{17} + \frac{3}{7} a^{16} + \frac{3}{7} a^{15} + \frac{3}{7} a^{14} - \frac{1}{7} a^{12} + \frac{1}{7} a^{11} + \frac{2}{7} a^{10} - \frac{3}{7} a^{8} - \frac{3}{7} a^{7} + \frac{2}{7} a^{6} + \frac{1}{7} a^{5} + \frac{1}{7} a^{4} - \frac{1}{7} a^{3} - \frac{2}{7} a^{2} + \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{21} - \frac{2}{7} a^{19} - \frac{2}{7} a^{18} - \frac{2}{7} a^{17} + \frac{2}{7} a^{16} + \frac{2}{7} a^{15} - \frac{1}{7} a^{14} - \frac{1}{7} a^{13} - \frac{1}{7} a^{12} - \frac{3}{7} a^{11} - \frac{3}{7} a^{10} - \frac{3}{7} a^{9} - \frac{2}{7} a^{8} + \frac{3}{7} a^{7} - \frac{2}{7} a^{6} + \frac{3}{7} a^{5} + \frac{1}{7} a^{4} + \frac{3}{7} a^{3} - \frac{1}{7} a^{2} + \frac{2}{7}$, $\frac{1}{77} a^{22} + \frac{5}{77} a^{21} - \frac{1}{77} a^{20} - \frac{1}{11} a^{19} + \frac{18}{77} a^{18} - \frac{5}{11} a^{17} - \frac{13}{77} a^{16} + \frac{3}{7} a^{15} + \frac{18}{77} a^{14} + \frac{8}{77} a^{13} + \frac{26}{77} a^{12} - \frac{10}{77} a^{11} + \frac{19}{77} a^{10} - \frac{3}{77} a^{9} - \frac{24}{77} a^{8} - \frac{32}{77} a^{7} + \frac{9}{77} a^{6} + \frac{31}{77} a^{5} - \frac{19}{77} a^{4} + \frac{34}{77} a^{3} - \frac{1}{11} a^{2} - \frac{37}{77} a - \frac{31}{77}$, $\frac{1}{104951} a^{23} + \frac{71}{104951} a^{22} - \frac{1002}{104951} a^{21} - \frac{39}{2233} a^{20} + \frac{20830}{104951} a^{19} + \frac{49729}{104951} a^{18} - \frac{5725}{14993} a^{17} + \frac{31}{203} a^{16} + \frac{39222}{104951} a^{15} - \frac{27162}{104951} a^{14} + \frac{50703}{104951} a^{13} - \frac{15762}{104951} a^{12} + \frac{758}{2233} a^{11} + \frac{20589}{104951} a^{10} + \frac{27256}{104951} a^{9} - \frac{601}{2233} a^{8} - \frac{60}{3619} a^{7} + \frac{8083}{104951} a^{6} + \frac{8363}{104951} a^{5} - \frac{14475}{104951} a^{4} - \frac{36725}{104951} a^{3} + \frac{13823}{104951} a^{2} - \frac{44097}{104951} a + \frac{476}{1363}$, $\frac{1}{260241248328479810066455605685221977} a^{24} + \frac{45878694745893157983527627333}{37177321189782830009493657955031711} a^{23} + \frac{90366593814727997836693737239242}{37177321189782830009493657955031711} a^{22} - \frac{899495744583495874688900697705814}{260241248328479810066455605685221977} a^{21} + \frac{16721829380183768782062789540870281}{260241248328479810066455605685221977} a^{20} + \frac{91948921908277394846368141899598254}{260241248328479810066455605685221977} a^{19} + \frac{18303263431204907845390813569404765}{260241248328479810066455605685221977} a^{18} - \frac{4425919379056206415810534802345126}{260241248328479810066455605685221977} a^{17} + \frac{5103653137356764700310380639263155}{260241248328479810066455605685221977} a^{16} - \frac{45611606276413347066478328716065449}{260241248328479810066455605685221977} a^{15} + \frac{35278568982405139987227812277898652}{260241248328479810066455605685221977} a^{14} + \frac{3329012248394719746886038207120792}{23658295302589073642405055062292907} a^{13} - \frac{13597290767286188993350744776160988}{260241248328479810066455605685221977} a^{12} - \frac{9283073417611264735438997977185914}{23658295302589073642405055062292907} a^{11} + \frac{110744345605584579566522737798191061}{260241248328479810066455605685221977} a^{10} - \frac{128075552204194243994745351876446882}{260241248328479810066455605685221977} a^{9} - \frac{4230976279085374431444956019285470}{23658295302589073642405055062292907} a^{8} + \frac{100861828667749445991170074840451216}{260241248328479810066455605685221977} a^{7} + \frac{24151931394826863155062146261867955}{260241248328479810066455605685221977} a^{6} - \frac{100027220903162140488394200681903256}{260241248328479810066455605685221977} a^{5} - \frac{18696058940493921653908981957450379}{260241248328479810066455605685221977} a^{4} - \frac{1477137359355527605319626794017188}{260241248328479810066455605685221977} a^{3} - \frac{62543993333397539512767840849598691}{260241248328479810066455605685221977} a^{2} - \frac{92197850307210230491479884051788848}{260241248328479810066455605685221977} a - \frac{72122110404046259596576556927021925}{260241248328479810066455605685221977}$

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:  \( 29321804577.651592 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{5}\cdot(2\pi)^{10}\cdot 29321804577.651592 \cdot 1}{2\sqrt{30047562514932432943626078739673704593849}}\approx 0.259540261271884$ (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

5.5.13845841.1, 5.1.124612569.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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
$3$3.5.0.1$x^{5} - x + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$61$61.5.4.1$x^{5} - 61$$5$$1$$4$$C_5$$[\ ]_{5}$
61.5.4.1$x^{5} - 61$$5$$1$$4$$C_5$$[\ ]_{5}$
61.5.4.1$x^{5} - 61$$5$$1$$4$$C_5$$[\ ]_{5}$
61.5.4.1$x^{5} - 61$$5$$1$$4$$C_5$$[\ ]_{5}$
61.5.4.1$x^{5} - 61$$5$$1$$4$$C_5$$[\ ]_{5}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
1.183.10t1.a.d$1$ $ 3 \cdot 61 $ 10.0.46584877058339283.1 $C_{10}$ (as 10T1) $0$ $-1$
* 1.61.5t1.a.c$1$ $ 61 $ 5.5.13845841.1 $C_5$ (as 5T1) $0$ $1$
1.183.10t1.a.a$1$ $ 3 \cdot 61 $ 10.0.46584877058339283.1 $C_{10}$ (as 10T1) $0$ $-1$
* 1.61.5t1.a.b$1$ $ 61 $ 5.5.13845841.1 $C_5$ (as 5T1) $0$ $1$
* 1.61.5t1.a.a$1$ $ 61 $ 5.5.13845841.1 $C_5$ (as 5T1) $0$ $1$
1.183.10t1.a.b$1$ $ 3 \cdot 61 $ 10.0.46584877058339283.1 $C_{10}$ (as 10T1) $0$ $-1$
* 1.61.5t1.a.d$1$ $ 61 $ 5.5.13845841.1 $C_5$ (as 5T1) $0$ $1$
1.183.10t1.a.c$1$ $ 3 \cdot 61 $ 10.0.46584877058339283.1 $C_{10}$ (as 10T1) $0$ $-1$
* 2.11163.10t6.a.a$2$ $ 3 \cdot 61^{2}$ 25.5.30047562514932432943626078739673704593849.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.11163.10t6.a.d$2$ $ 3 \cdot 61^{2}$ 25.5.30047562514932432943626078739673704593849.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.183.10t6.a.c$2$ $ 3 \cdot 61 $ 25.5.30047562514932432943626078739673704593849.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.183.10t6.a.a$2$ $ 3 \cdot 61 $ 25.5.30047562514932432943626078739673704593849.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.11163.5t2.a.a$2$ $ 3 \cdot 61^{2}$ 5.1.124612569.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.183.10t6.a.d$2$ $ 3 \cdot 61 $ 25.5.30047562514932432943626078739673704593849.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.183.10t6.a.b$2$ $ 3 \cdot 61 $ 25.5.30047562514932432943626078739673704593849.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.11163.10t6.a.c$2$ $ 3 \cdot 61^{2}$ 25.5.30047562514932432943626078739673704593849.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.11163.5t2.a.b$2$ $ 3 \cdot 61^{2}$ 5.1.124612569.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.11163.10t6.a.b$2$ $ 3 \cdot 61^{2}$ 25.5.30047562514932432943626078739673704593849.1 $C_5\times D_5$ (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 *.