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Properties

Label 20.2.30988918815...0000.2
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{4}\cdot 5^{14}\cdot 19^{4}\cdot 29^{7}\cdot 109^{4}$
Root discriminant $53.03$
Ramified primes $2, 5, 19, 29, 109$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 20T955

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![754525, 607525, -955925, -647150, 301880, 143415, 82710, 27270, -43266, -2145, -1742, -2650, 153, -306, 715, -161, 87, -63, 29, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 29*x^18 - 63*x^17 + 87*x^16 - 161*x^15 + 715*x^14 - 306*x^13 + 153*x^12 - 2650*x^11 - 1742*x^10 - 2145*x^9 - 43266*x^8 + 27270*x^7 + 82710*x^6 + 143415*x^5 + 301880*x^4 - 647150*x^3 - 955925*x^2 + 607525*x + 754525)
 
gp: K = bnfinit(x^20 - 7*x^19 + 29*x^18 - 63*x^17 + 87*x^16 - 161*x^15 + 715*x^14 - 306*x^13 + 153*x^12 - 2650*x^11 - 1742*x^10 - 2145*x^9 - 43266*x^8 + 27270*x^7 + 82710*x^6 + 143415*x^5 + 301880*x^4 - 647150*x^3 - 955925*x^2 + 607525*x + 754525, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 29 x^{18} - 63 x^{17} + 87 x^{16} - 161 x^{15} + 715 x^{14} - 306 x^{13} + 153 x^{12} - 2650 x^{11} - 1742 x^{10} - 2145 x^{9} - 43266 x^{8} + 27270 x^{7} + 82710 x^{6} + 143415 x^{5} + 301880 x^{4} - 647150 x^{3} - 955925 x^{2} + 607525 x + 754525 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-30988918815000698609123925781250000=-\,2^{4}\cdot 5^{14}\cdot 19^{4}\cdot 29^{7}\cdot 109^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.03$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5} a^{15} + \frac{1}{5} a^{14} + \frac{2}{5} a^{13} - \frac{2}{5} a^{12} + \frac{1}{5} a^{11} + \frac{2}{5} a^{10} + \frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4}$, $\frac{1}{5} a^{16} + \frac{1}{5} a^{14} + \frac{1}{5} a^{13} - \frac{2}{5} a^{12} + \frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4}$, $\frac{1}{5} a^{17} + \frac{1}{5} a^{13} - \frac{2}{5} a^{12} - \frac{2}{5} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4}$, $\frac{1}{15} a^{18} + \frac{1}{15} a^{17} - \frac{1}{15} a^{16} - \frac{1}{15} a^{15} - \frac{2}{5} a^{14} + \frac{2}{5} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{15} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{7}{15} a^{7} + \frac{2}{5} a^{6} - \frac{1}{3} a^{5} - \frac{7}{15} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{1115634659969845945151713919163905755357651852655572660725} a^{19} - \frac{6030953022625758651624778333345158448025695570260179107}{1115634659969845945151713919163905755357651852655572660725} a^{18} + \frac{36310083785787836696488287624800291316570068885695378513}{371878219989948648383904639721301918452550617551857553575} a^{17} - \frac{3671829585725707591979500853362547464226042417891580743}{1115634659969845945151713919163905755357651852655572660725} a^{16} + \frac{11852336333127184286425490460427853268662572658376744617}{1115634659969845945151713919163905755357651852655572660725} a^{15} - \frac{36173252235755955790339864564812065675265105519561068952}{371878219989948648383904639721301918452550617551857553575} a^{14} + \frac{109341558978465708445199627811611314676726966275700355632}{223126931993969189030342783832781151071530370531114532145} a^{13} - \frac{77952434296782252108229031296894530581051365937395421172}{371878219989948648383904639721301918452550617551857553575} a^{12} - \frac{236492114422153775601552411312143799474570848373185693702}{1115634659969845945151713919163905755357651852655572660725} a^{11} - \frac{8622480662423021954764103534157012959816741782486246138}{223126931993969189030342783832781151071530370531114532145} a^{10} - \frac{120753036674213018755601637019692595479762877409864499149}{371878219989948648383904639721301918452550617551857553575} a^{9} + \frac{1969393624933655582667732902606943026806919925653784349}{223126931993969189030342783832781151071530370531114532145} a^{8} - \frac{387911626006158677803245501935363388342732955317259163846}{1115634659969845945151713919163905755357651852655572660725} a^{7} + \frac{19924865548903663574698548234761157069393563343565997809}{44625386398793837806068556766556230214306074106222906429} a^{6} + \frac{17685340219642102897135681149196804415860451300481635801}{74375643997989729676780927944260383690510123510371510715} a^{5} + \frac{48924722881772040297557133715992606152679765864364307717}{223126931993969189030342783832781151071530370531114532145} a^{4} + \frac{11622452186486357252992902395209195112040510508546201073}{223126931993969189030342783832781151071530370531114532145} a^{3} - \frac{5037525257312279864707247633743626128061077268996215659}{14875128799597945935356185588852076738102024702074302143} a^{2} - \frac{1549645889591873072186362032734735347713458410389471987}{14875128799597945935356185588852076738102024702074302143} a + \frac{6306931668507185283880244903254700011081315266482228957}{44625386398793837806068556766556230214306074106222906429}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $10$
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:  \( 858876864.343 \) (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 $16{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ R $20$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ $16{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_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}$
19.5.0.1$x^{5} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
19.5.0.1$x^{5} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
$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}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{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}$