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Properties

Label 20.4.11647559140...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{4}\cdot 5^{14}\cdot 19^{4}\cdot 29^{6}\cdot 109^{5}$
Root discriminant $56.66$
Ramified primes $2, 5, 19, 29, 109$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T955

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![59049, -78732, -52488, 32805, -23328, 3402, 15876, -9612, 2376, 3261, -1790, 1087, 264, -356, 196, 14, -32, 15, -8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 8*x^18 + 15*x^17 - 32*x^16 + 14*x^15 + 196*x^14 - 356*x^13 + 264*x^12 + 1087*x^11 - 1790*x^10 + 3261*x^9 + 2376*x^8 - 9612*x^7 + 15876*x^6 + 3402*x^5 - 23328*x^4 + 32805*x^3 - 52488*x^2 - 78732*x + 59049)
 
gp: K = bnfinit(x^20 - 4*x^19 - 8*x^18 + 15*x^17 - 32*x^16 + 14*x^15 + 196*x^14 - 356*x^13 + 264*x^12 + 1087*x^11 - 1790*x^10 + 3261*x^9 + 2376*x^8 - 9612*x^7 + 15876*x^6 + 3402*x^5 - 23328*x^4 + 32805*x^3 - 52488*x^2 - 78732*x + 59049, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 8 x^{18} + 15 x^{17} - 32 x^{16} + 14 x^{15} + 196 x^{14} - 356 x^{13} + 264 x^{12} + 1087 x^{11} - 1790 x^{10} + 3261 x^{9} + 2376 x^{8} - 9612 x^{7} + 15876 x^{6} + 3402 x^{5} - 23328 x^{4} + 32805 x^{3} - 52488 x^{2} - 78732 x + 59049 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(116475591408106074082569238281250000=2^{4}\cdot 5^{14}\cdot 19^{4}\cdot 29^{6}\cdot 109^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.66$
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:  $2, 5, 19, 29, 109$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{15} a^{11} - \frac{1}{15} a^{10} + \frac{7}{15} a^{9} - \frac{1}{5} a^{8} - \frac{1}{3} a^{7} - \frac{4}{15} a^{6} - \frac{1}{3} a^{5} - \frac{2}{15} a^{4} + \frac{1}{15} a^{2} + \frac{1}{15} a + \frac{1}{5}$, $\frac{1}{45} a^{12} - \frac{1}{45} a^{11} - \frac{2}{45} a^{10} + \frac{4}{45} a^{8} - \frac{1}{45} a^{7} + \frac{22}{45} a^{6} + \frac{7}{45} a^{5} + \frac{2}{15} a^{4} - \frac{2}{45} a^{3} + \frac{4}{45} a^{2} + \frac{2}{15} a + \frac{2}{5}$, $\frac{1}{135} a^{13} - \frac{1}{135} a^{12} - \frac{2}{135} a^{11} + \frac{4}{135} a^{9} - \frac{1}{135} a^{8} + \frac{67}{135} a^{7} + \frac{52}{135} a^{6} + \frac{17}{45} a^{5} - \frac{47}{135} a^{4} - \frac{41}{135} a^{3} - \frac{13}{45} a^{2} - \frac{1}{5} a$, $\frac{1}{405} a^{14} - \frac{1}{405} a^{13} - \frac{2}{405} a^{12} + \frac{31}{405} a^{10} - \frac{11}{81} a^{9} - \frac{19}{81} a^{8} - \frac{2}{405} a^{7} - \frac{2}{27} a^{6} - \frac{74}{405} a^{5} - \frac{14}{405} a^{4} - \frac{8}{27} a^{3} + \frac{7}{15} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{1215} a^{15} - \frac{1}{1215} a^{14} - \frac{2}{1215} a^{13} + \frac{31}{1215} a^{11} + \frac{26}{1215} a^{10} + \frac{148}{1215} a^{9} + \frac{322}{1215} a^{8} + \frac{71}{405} a^{7} - \frac{317}{1215} a^{6} - \frac{100}{243} a^{5} - \frac{13}{405} a^{4} - \frac{2}{45} a^{3} - \frac{1}{15} a^{2} - \frac{1}{15} a + \frac{1}{5}$, $\frac{1}{3645} a^{16} - \frac{1}{3645} a^{15} - \frac{2}{3645} a^{14} + \frac{31}{3645} a^{12} + \frac{26}{3645} a^{11} - \frac{338}{3645} a^{10} - \frac{1136}{3645} a^{9} + \frac{233}{1215} a^{8} - \frac{112}{729} a^{7} + \frac{958}{3645} a^{6} + \frac{554}{1215} a^{5} - \frac{13}{27} a^{4} + \frac{17}{45} a^{3} + \frac{11}{45} a^{2} - \frac{2}{5}$, $\frac{1}{10935} a^{17} - \frac{1}{10935} a^{16} - \frac{2}{10935} a^{15} + \frac{31}{10935} a^{13} + \frac{26}{10935} a^{12} - \frac{338}{10935} a^{11} - \frac{407}{10935} a^{10} - \frac{1468}{3645} a^{9} - \frac{4934}{10935} a^{8} + \frac{629}{2187} a^{7} - \frac{35}{729} a^{6} - \frac{92}{405} a^{5} + \frac{26}{135} a^{4} - \frac{16}{135} a^{3} - \frac{7}{15} a^{2} - \frac{4}{15} a + \frac{1}{5}$, $\frac{1}{77773765950} a^{18} - \frac{1338683}{38886882975} a^{17} - \frac{2352089}{77773765950} a^{16} + \frac{295309}{8641529550} a^{15} - \frac{72196997}{77773765950} a^{14} + \frac{8535409}{15554753190} a^{13} - \frac{255923573}{77773765950} a^{12} + \frac{336949709}{15554753190} a^{11} - \frac{1332982519}{25924588650} a^{10} + \frac{8416599803}{38886882975} a^{9} - \frac{35095449641}{77773765950} a^{8} + \frac{962010853}{5184917730} a^{7} + \frac{1095426727}{2880509850} a^{6} - \frac{5214043}{576101970} a^{5} - \frac{418108141}{960169950} a^{4} - \frac{49499599}{106685550} a^{3} + \frac{5538077}{11853950} a^{2} - \frac{153288}{5926975} a + \frac{2370793}{11853950}$, $\frac{1}{233321297850} a^{19} - \frac{1}{233321297850} a^{18} - \frac{919739}{233321297850} a^{17} - \frac{340918}{12962294325} a^{16} + \frac{402419}{116660648925} a^{15} - \frac{3253033}{23332129785} a^{14} - \frac{389566069}{116660648925} a^{13} + \frac{178556128}{23332129785} a^{12} - \frac{198463367}{38886882975} a^{11} - \frac{12264462209}{233321297850} a^{10} - \frac{4736793911}{233321297850} a^{9} - \frac{2035446694}{7777376595} a^{8} + \frac{5666946718}{12962294325} a^{7} + \frac{100563097}{288050985} a^{6} - \frac{77056957}{160028325} a^{5} - \frac{217353241}{480084975} a^{4} - \frac{39336311}{160028325} a^{3} - \frac{7343243}{106685550} a^{2} + \frac{6173933}{35561850} a - \frac{887001}{2370790}$

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:  $11$
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:  \( 7615773802.29 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T955:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 819200
The 275 conjugacy class representatives for t20n955 are not computed
Character table for t20n955 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.8172298511640625.1

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ R $20$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ $16{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

In the table, R denotes a ramified prime. 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
$2$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.10.0.1$x^{10} + x^{2} - 2 x + 14$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$109$109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.4.2.1$x^{4} + 1199 x^{2} + 427716$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
109.4.3.4$x^{4} + 23544$$4$$1$$3$$C_4$$[\ ]_{4}$
109.4.0.1$x^{4} - x + 30$$1$$4$$0$$C_4$$[\ ]^{4}$