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Properties

Label 20.6.59313783170...0256.1
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{20}\cdot 3^{19}\cdot 13^{5}\cdot 107^{4}$
Root discriminant $27.46$
Ramified primes $2, 3, 13, 107$
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![9, -36, -108, 216, 471, -342, -753, -78, 463, 744, -294, -600, 447, -12, -217, 162, -15, -24, 21, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 21*x^18 - 24*x^17 - 15*x^16 + 162*x^15 - 217*x^14 - 12*x^13 + 447*x^12 - 600*x^11 - 294*x^10 + 744*x^9 + 463*x^8 - 78*x^7 - 753*x^6 - 342*x^5 + 471*x^4 + 216*x^3 - 108*x^2 - 36*x + 9)
 
gp: K = bnfinit(x^20 - 6*x^19 + 21*x^18 - 24*x^17 - 15*x^16 + 162*x^15 - 217*x^14 - 12*x^13 + 447*x^12 - 600*x^11 - 294*x^10 + 744*x^9 + 463*x^8 - 78*x^7 - 753*x^6 - 342*x^5 + 471*x^4 + 216*x^3 - 108*x^2 - 36*x + 9, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 21 x^{18} - 24 x^{17} - 15 x^{16} + 162 x^{15} - 217 x^{14} - 12 x^{13} + 447 x^{12} - 600 x^{11} - 294 x^{10} + 744 x^{9} + 463 x^{8} - 78 x^{7} - 753 x^{6} - 342 x^{5} + 471 x^{4} + 216 x^{3} - 108 x^{2} - 36 x + 9 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-59313783170950766384309600256=-\,2^{20}\cdot 3^{19}\cdot 13^{5}\cdot 107^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.46$
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:  $2, 3, 13, 107$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{10} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{11} + \frac{1}{3} a^{5}$, $\frac{1}{21} a^{18} - \frac{2}{21} a^{17} + \frac{1}{7} a^{16} - \frac{2}{7} a^{15} - \frac{2}{7} a^{14} + \frac{3}{7} a^{13} + \frac{5}{21} a^{12} + \frac{2}{21} a^{11} + \frac{2}{7} a^{10} + \frac{2}{7} a^{9} + \frac{2}{7} a^{8} - \frac{2}{7} a^{7} + \frac{1}{21} a^{6} + \frac{1}{3} a^{5} - \frac{2}{7} a^{3} + \frac{2}{7} a^{2} + \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{2651853972348201847313325} a^{19} - \frac{59008824760945592221828}{2651853972348201847313325} a^{18} - \frac{191127601955687934340163}{2651853972348201847313325} a^{17} + \frac{8323158297792256919904}{883951324116067282437775} a^{16} + \frac{29238483655313496636407}{883951324116067282437775} a^{15} + \frac{8676240802506337751370}{35358052964642691297511} a^{14} - \frac{378687816635186545084867}{2651853972348201847313325} a^{13} - \frac{191270731158943394968538}{2651853972348201847313325} a^{12} + \frac{533400777121005867845483}{2651853972348201847313325} a^{11} + \frac{235485356900829883948283}{883951324116067282437775} a^{10} + \frac{176653501160090552246076}{883951324116067282437775} a^{9} - \frac{346172895758925621212599}{883951324116067282437775} a^{8} + \frac{1161558899083523494926397}{2651853972348201847313325} a^{7} + \frac{938114315131001785974788}{2651853972348201847313325} a^{6} - \frac{16308378304003393760102}{378836281764028835330475} a^{5} + \frac{124642688676538949611497}{883951324116067282437775} a^{4} - \frac{53806444183815714898602}{883951324116067282437775} a^{3} - \frac{355095347133985566155459}{883951324116067282437775} a^{2} + \frac{24015341116770066163591}{126278760588009611776825} a + \frac{199603579225690228919599}{883951324116067282437775}$

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:  $12$
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:  \( 7112380.29553 \) (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

\(\Q(\sqrt{3}) \), 10.6.38998285028352.1

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 R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ R $20$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}$ $20$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
$3$3.8.7.2$x^{8} - 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
3.12.12.28$x^{12} + 12 x^{11} - 3 x^{10} + 3 x^{9} + 3 x^{8} + 6 x^{7} + 12 x^{6} + 9 x^{5} + 9 x^{4} + 9 x + 9$$6$$2$$12$12T34$[5/4, 5/4]_{4}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.6.5.6$x^{6} + 416$$6$$1$$5$$C_6$$[\ ]_{6}$
$107$$\Q_{107}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{107}$$x + 3$$1$$1$$0$Trivial$[\ ]$
107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
107.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
107.6.4.1$x^{6} + 1498 x^{3} + 1431125$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$