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Properties

Label 20.8.86829841664...0000.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{16}\cdot 5^{11}\cdot 19^{4}\cdot 461^{5}$
Root discriminant $35.23$
Ramified primes $2, 5, 19, 461$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1023

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![29, 311, -820, -2021, 1708, 4241, 1294, -461, -991, -2612, -2276, -718, -525, -529, 12, 35, -54, 17, 6, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 6*x^18 + 17*x^17 - 54*x^16 + 35*x^15 + 12*x^14 - 529*x^13 - 525*x^12 - 718*x^11 - 2276*x^10 - 2612*x^9 - 991*x^8 - 461*x^7 + 1294*x^6 + 4241*x^5 + 1708*x^4 - 2021*x^3 - 820*x^2 + 311*x + 29)
 
gp: K = bnfinit(x^20 - 3*x^19 + 6*x^18 + 17*x^17 - 54*x^16 + 35*x^15 + 12*x^14 - 529*x^13 - 525*x^12 - 718*x^11 - 2276*x^10 - 2612*x^9 - 991*x^8 - 461*x^7 + 1294*x^6 + 4241*x^5 + 1708*x^4 - 2021*x^3 - 820*x^2 + 311*x + 29, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 6 x^{18} + 17 x^{17} - 54 x^{16} + 35 x^{15} + 12 x^{14} - 529 x^{13} - 525 x^{12} - 718 x^{11} - 2276 x^{10} - 2612 x^{9} - 991 x^{8} - 461 x^{7} + 1294 x^{6} + 4241 x^{5} + 1708 x^{4} - 2021 x^{3} - 820 x^{2} + 311 x + 29 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8682984166419406787200000000000=2^{16}\cdot 5^{11}\cdot 19^{4}\cdot 461^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.23$
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:  $2, 5, 19, 461$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{38} a^{17} - \frac{3}{38} a^{16} + \frac{11}{38} a^{14} - \frac{1}{38} a^{13} - \frac{6}{19} a^{12} - \frac{9}{19} a^{11} + \frac{2}{19} a^{10} + \frac{2}{19} a^{9} - \frac{3}{19} a^{8} - \frac{1}{19} a^{7} - \frac{6}{19} a^{6} + \frac{1}{38} a^{5} + \frac{1}{38} a^{4} + \frac{9}{19} a^{3} - \frac{7}{38} a^{2} - \frac{1}{38} a + \frac{1}{19}$, $\frac{1}{38} a^{18} - \frac{9}{38} a^{16} - \frac{4}{19} a^{15} - \frac{3}{19} a^{14} - \frac{15}{38} a^{13} + \frac{3}{38} a^{12} - \frac{6}{19} a^{11} + \frac{8}{19} a^{10} + \frac{3}{19} a^{9} + \frac{9}{19} a^{8} - \frac{9}{19} a^{7} + \frac{3}{38} a^{6} + \frac{2}{19} a^{5} - \frac{17}{38} a^{4} - \frac{5}{19} a^{3} + \frac{8}{19} a^{2} - \frac{1}{38} a - \frac{13}{38}$, $\frac{1}{81671596320586932758884330237334} a^{19} - \frac{122643750240492224006606507306}{40835798160293466379442165118667} a^{18} - \frac{220490173557564667267094247047}{40835798160293466379442165118667} a^{17} - \frac{822642519410202125985727735264}{40835798160293466379442165118667} a^{16} + \frac{9836285073983356971122390454160}{40835798160293466379442165118667} a^{15} + \frac{2448586472726780493559326995697}{40835798160293466379442165118667} a^{14} - \frac{32435720158346830979900700689767}{81671596320586932758884330237334} a^{13} + \frac{18631268145473581143557230983486}{40835798160293466379442165118667} a^{12} - \frac{3001187580661903907089433548446}{40835798160293466379442165118667} a^{11} - \frac{8101936064278323438387072964963}{40835798160293466379442165118667} a^{10} - \frac{7886420890639734072200104898824}{40835798160293466379442165118667} a^{9} - \frac{10781966269698423497147130807397}{40835798160293466379442165118667} a^{8} - \frac{20452717486912607517641655313653}{81671596320586932758884330237334} a^{7} - \frac{19574253542451061939930979374512}{40835798160293466379442165118667} a^{6} - \frac{18898826673799416929758525029536}{40835798160293466379442165118667} a^{5} + \frac{20225586448368834735864015594189}{40835798160293466379442165118667} a^{4} + \frac{13341887986242081079898138642363}{40835798160293466379442165118667} a^{3} + \frac{7174328741102316569616519370844}{40835798160293466379442165118667} a^{2} - \frac{30973641231751708663233076253229}{81671596320586932758884330237334} a - \frac{9427850122713709269899448019150}{40835798160293466379442165118667}$

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:  $13$
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:  \( 168929733.41 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1023:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 324 conjugacy class representatives for t20n1023 are not computed
Character table for t20n1023 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.61376064800000.1

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

Sibling fields

Degree 20 sibling: 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $20$ R ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.10.0.1$x^{10} + x^{2} - 2 x + 14$$1$$10$$0$$C_{10}$$[\ ]^{10}$
461Data not computed