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Properties

Label 20.12.2729663703...4832.2
Degree $20$
Signature $[12, 4]$
Discriminant $2^{34}\cdot 7^{9}\cdot 13^{14}$
Root discriminant $46.97$
Ramified primes $2, 7, 13$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T633

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-712, -2384, 4744, 24696, 12185, -50132, -68428, -6310, 44530, 31066, -3933, -13146, -4020, 1824, 1267, -38, -94, 6, -9, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 9*x^18 + 6*x^17 - 94*x^16 - 38*x^15 + 1267*x^14 + 1824*x^13 - 4020*x^12 - 13146*x^11 - 3933*x^10 + 31066*x^9 + 44530*x^8 - 6310*x^7 - 68428*x^6 - 50132*x^5 + 12185*x^4 + 24696*x^3 + 4744*x^2 - 2384*x - 712)
 
gp: K = bnfinit(x^20 - 2*x^19 - 9*x^18 + 6*x^17 - 94*x^16 - 38*x^15 + 1267*x^14 + 1824*x^13 - 4020*x^12 - 13146*x^11 - 3933*x^10 + 31066*x^9 + 44530*x^8 - 6310*x^7 - 68428*x^6 - 50132*x^5 + 12185*x^4 + 24696*x^3 + 4744*x^2 - 2384*x - 712, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 9 x^{18} + 6 x^{17} - 94 x^{16} - 38 x^{15} + 1267 x^{14} + 1824 x^{13} - 4020 x^{12} - 13146 x^{11} - 3933 x^{10} + 31066 x^{9} + 44530 x^{8} - 6310 x^{7} - 68428 x^{6} - 50132 x^{5} + 12185 x^{4} + 24696 x^{3} + 4744 x^{2} - 2384 x - 712 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2729663703817172900595808790904832=2^{34}\cdot 7^{9}\cdot 13^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.97$
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:  $2, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{13} a^{16} - \frac{4}{13} a^{14} + \frac{5}{13} a^{13} + \frac{4}{13} a^{12} + \frac{6}{13} a^{11} + \frac{5}{13} a^{10} + \frac{1}{13} a^{9} + \frac{4}{13} a^{8} + \frac{4}{13} a^{7} - \frac{4}{13} a^{6} + \frac{1}{13} a^{5} + \frac{5}{13} a^{4} - \frac{2}{13} a^{3} + \frac{3}{13} a^{2} + \frac{5}{13} a - \frac{4}{13}$, $\frac{1}{91} a^{17} - \frac{2}{91} a^{16} + \frac{22}{91} a^{15} - \frac{2}{7} a^{14} - \frac{32}{91} a^{13} - \frac{2}{91} a^{12} + \frac{6}{91} a^{11} + \frac{4}{91} a^{10} - \frac{37}{91} a^{9} - \frac{4}{91} a^{8} - \frac{25}{91} a^{7} - \frac{30}{91} a^{6} - \frac{10}{91} a^{5} - \frac{38}{91} a^{4} + \frac{33}{91} a^{3} - \frac{27}{91} a^{2} - \frac{2}{13} a + \frac{34}{91}$, $\frac{1}{5278} a^{18} + \frac{12}{2639} a^{17} - \frac{37}{5278} a^{16} - \frac{2}{29} a^{15} - \frac{431}{2639} a^{14} + \frac{1067}{2639} a^{13} + \frac{1655}{5278} a^{12} - \frac{1124}{2639} a^{11} + \frac{1290}{2639} a^{10} - \frac{89}{377} a^{9} - \frac{703}{5278} a^{8} - \frac{718}{2639} a^{7} - \frac{472}{2639} a^{6} - \frac{1199}{2639} a^{5} - \frac{1041}{2639} a^{4} - \frac{13}{203} a^{3} + \frac{355}{5278} a^{2} + \frac{682}{2639} a + \frac{1002}{2639}$, $\frac{1}{521966039354741044972086018944012} a^{19} + \frac{2043951281116171623376093062}{130491509838685261243021504736003} a^{18} - \frac{305361841696047020788583405065}{521966039354741044972086018944012} a^{17} + \frac{72771375831982035462299229124}{5673543906029793967087891510261} a^{16} + \frac{43164126171270974793406058871587}{260983019677370522486043009472006} a^{15} - \frac{40221642914144656766132414727195}{260983019677370522486043009472006} a^{14} - \frac{6005685153042937516234551182365}{40151233796518541920929693764924} a^{13} - \frac{115253809317509061550979067660007}{260983019677370522486043009472006} a^{12} + \frac{27171743203459366502833854859579}{130491509838685261243021504736003} a^{11} + \frac{127861662891896351697801272391477}{260983019677370522486043009472006} a^{10} + \frac{34591861084383703569815299522635}{521966039354741044972086018944012} a^{9} + \frac{40203377290781475417400437510014}{130491509838685261243021504736003} a^{8} + \frac{35449059597168919606376535223577}{260983019677370522486043009472006} a^{7} + \frac{12153261843380804927046358855941}{37283288525338646069434715638858} a^{6} + \frac{151389764823305421220904031382}{810506272289970566726841644323} a^{5} + \frac{8039411062613713436909187400564}{130491509838685261243021504736003} a^{4} - \frac{227244554063675631058621474512803}{521966039354741044972086018944012} a^{3} - \frac{6491326628508599508600506547653}{37283288525338646069434715638858} a^{2} - \frac{16515667148226062204892169618133}{130491509838685261243021504736003} a + \frac{43541007461700355793218401650692}{130491509838685261243021504736003}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $15$
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:  \( 4608214386.88 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T633:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40960
The 124 conjugacy class representatives for t20n633 are not computed
Character table for t20n633 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 5.5.6889792.1, 10.10.379753870426112.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ R ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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
2Data not computed
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$