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Properties

Label 20.2.21155249330...2464.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{20}\cdot 3^{18}\cdot 13^{5}\cdot 107^{5}$
Root discriminant $32.83$
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![-423, 252, -3366, 2040, -2151, -384, 2038, -636, 791, -814, 15, 1168, -93, -222, 102, 100, 18, -16, -7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^18 - 16*x^17 + 18*x^16 + 100*x^15 + 102*x^14 - 222*x^13 - 93*x^12 + 1168*x^11 + 15*x^10 - 814*x^9 + 791*x^8 - 636*x^7 + 2038*x^6 - 384*x^5 - 2151*x^4 + 2040*x^3 - 3366*x^2 + 252*x - 423)
 
gp: K = bnfinit(x^20 - 7*x^18 - 16*x^17 + 18*x^16 + 100*x^15 + 102*x^14 - 222*x^13 - 93*x^12 + 1168*x^11 + 15*x^10 - 814*x^9 + 791*x^8 - 636*x^7 + 2038*x^6 - 384*x^5 - 2151*x^4 + 2040*x^3 - 3366*x^2 + 252*x - 423, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{18} - 16 x^{17} + 18 x^{16} + 100 x^{15} + 102 x^{14} - 222 x^{13} - 93 x^{12} + 1168 x^{11} + 15 x^{10} - 814 x^{9} + 791 x^{8} - 636 x^{7} + 2038 x^{6} - 384 x^{5} - 2151 x^{4} + 2040 x^{3} - 3366 x^{2} + 252 x - 423 \)

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:  \(-2115524933097244001040375742464=-\,2^{20}\cdot 3^{18}\cdot 13^{5}\cdot 107^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{9} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{10} + \frac{1}{3} a^{4}$, $\frac{1}{9} a^{17} - \frac{1}{9} a^{15} + \frac{1}{9} a^{14} - \frac{1}{9} a^{12} - \frac{1}{9} a^{11} + \frac{1}{3} a^{10} - \frac{2}{9} a^{9} - \frac{4}{9} a^{8} + \frac{1}{3} a^{7} - \frac{2}{9} a^{6} - \frac{1}{3} a^{5}$, $\frac{1}{2691} a^{18} + \frac{128}{2691} a^{17} + \frac{35}{2691} a^{16} - \frac{292}{2691} a^{15} - \frac{265}{2691} a^{14} + \frac{389}{2691} a^{13} + \frac{58}{897} a^{12} - \frac{374}{2691} a^{11} - \frac{80}{207} a^{10} - \frac{44}{2691} a^{9} + \frac{883}{2691} a^{8} + \frac{1234}{2691} a^{7} - \frac{571}{2691} a^{6} + \frac{9}{23} a^{5} - \frac{103}{299} a^{4} + \frac{61}{897} a^{3} + \frac{27}{299} a^{2} + \frac{96}{299} a + \frac{44}{299}$, $\frac{1}{59210767130255734340761365000933} a^{19} - \frac{9696009861696206863852026910}{59210767130255734340761365000933} a^{18} + \frac{1858305241601566957266330915640}{59210767130255734340761365000933} a^{17} + \frac{6791553137612591142428102925824}{59210767130255734340761365000933} a^{16} + \frac{2260043719164872560577987265026}{19736922376751911446920455000311} a^{15} - \frac{73979718373195793461582254631}{2574381179576336275685276739171} a^{14} - \frac{1689043882664952585162074050150}{19736922376751911446920455000311} a^{13} + \frac{3156920049223676314779249349310}{59210767130255734340761365000933} a^{12} - \frac{4972571665805210420904516624982}{59210767130255734340761365000933} a^{11} + \frac{1788734017780054679595409557235}{59210767130255734340761365000933} a^{10} + \frac{3196943054070279985440838441111}{19736922376751911446920455000311} a^{9} + \frac{8720213329609209413401591278722}{59210767130255734340761365000933} a^{8} + \frac{12003539791565074312876186373084}{59210767130255734340761365000933} a^{7} - \frac{1916536986706984578817087328845}{59210767130255734340761365000933} a^{6} - \frac{2902796841087838332070822554151}{6578974125583970482306818333437} a^{5} - \frac{1435922441676965450676063625867}{6578974125583970482306818333437} a^{4} + \frac{63595209503238430323355905652}{506074932737228498638986025649} a^{3} + \frac{2396488441570079365739875210653}{6578974125583970482306818333437} a^{2} + \frac{2241720853471721155977605415538}{6578974125583970482306818333437} a + \frac{1293684687701732331358198858929}{6578974125583970482306818333437}$

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:  $10$
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:  \( 62306146.5873 \) (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 ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ R $20$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.6.6.7$x^{6} + 3 x + 6$$6$$1$$6$$C_3^2:D_4$$[5/4, 5/4]_{4}^{2}$
3.6.6.7$x^{6} + 3 x + 6$$6$$1$$6$$C_3^2:D_4$$[5/4, 5/4]_{4}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{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}$
13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
107Data not computed