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Properties

Label 20.4.37505108549...6256.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{54}\cdot 113^{6}$
Root discriminant $26.84$
Ramified primes $2, 113$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T201

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 16, 6, -116, -162, 80, 386, 652, -476, 192, -673, -94, 215, 80, -184, 0, 62, 16, -11, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 11*x^18 + 16*x^17 + 62*x^16 - 184*x^14 + 80*x^13 + 215*x^12 - 94*x^11 - 673*x^10 + 192*x^9 - 476*x^8 + 652*x^7 + 386*x^6 + 80*x^5 - 162*x^4 - 116*x^3 + 6*x^2 + 16*x + 4)
 
gp: K = bnfinit(x^20 - 2*x^19 - 11*x^18 + 16*x^17 + 62*x^16 - 184*x^14 + 80*x^13 + 215*x^12 - 94*x^11 - 673*x^10 + 192*x^9 - 476*x^8 + 652*x^7 + 386*x^6 + 80*x^5 - 162*x^4 - 116*x^3 + 6*x^2 + 16*x + 4, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 11 x^{18} + 16 x^{17} + 62 x^{16} - 184 x^{14} + 80 x^{13} + 215 x^{12} - 94 x^{11} - 673 x^{10} + 192 x^{9} - 476 x^{8} + 652 x^{7} + 386 x^{6} + 80 x^{5} - 162 x^{4} - 116 x^{3} + 6 x^{2} + 16 x + 4 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(37505108549012973893510496256=2^{54}\cdot 113^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.84$
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:  $2, 113$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11}$, $\frac{1}{8} a^{16} - \frac{1}{4} a^{15} + \frac{1}{8} a^{14} + \frac{1}{8} a^{12} - \frac{1}{4} a^{11} - \frac{1}{8} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{17} + \frac{1}{8} a^{15} - \frac{1}{4} a^{14} + \frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{18} - \frac{1}{4} a^{12} - \frac{1}{8} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{55548126249255530046314928} a^{19} + \frac{6884453755074717624647}{293905429890240899715952} a^{18} - \frac{99046196163130924003004}{3471757890578470627894683} a^{17} - \frac{2764766843185852586393}{210409569125967916842102} a^{16} - \frac{857673738834630659492092}{3471757890578470627894683} a^{15} + \frac{81742839699869508921037}{13887031562313882511578732} a^{14} + \frac{4897876855287853378905083}{27774063124627765023157464} a^{13} + \frac{546564191184995464671713}{3967723303518252146165352} a^{12} - \frac{23973046385372852347786313}{55548126249255530046314928} a^{11} + \frac{3337525089776604584751655}{7935446607036504292330704} a^{10} + \frac{1088827042838196141720505}{2524914829511615002105224} a^{9} - \frac{6562255941912570093120545}{27774063124627765023157464} a^{8} - \frac{6807862527061813081891445}{27774063124627765023157464} a^{7} - \frac{103206150077441245275697}{259570683407736121711752} a^{6} + \frac{715188728609968428201565}{1543003506923764723508748} a^{5} - \frac{4362518694282877014227683}{13887031562313882511578732} a^{4} - \frac{120289751608336895341937}{360702118501659286015032} a^{3} + \frac{721248636215373419963611}{9258021041542588341052488} a^{2} - \frac{29771767119158890577383}{1543003506923764723508748} a + \frac{3711999688782818614107097}{13887031562313882511578732}$

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:  $11$
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:  \( 39419063.9653 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T201:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1440
The 13 conjugacy class representatives for t20n201
Character table for t20n201

Intermediate fields

\(\Q(\sqrt{2}) \), 10.2.24207794634752.1

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

Sibling fields

Degree 10 sibling: data not computed
Degree 12 sibling: data not computed
Degree 20 siblings: data not computed
Degree 24 sibling: data not computed
Degree 30 sibling: data not computed
Degree 36 sibling: data not computed
Degree 40 siblings: data not computed
Degree 45 sibling: 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.8.24.4$x^{8} + 14 x^{4} + 8 x^{3} + 12 x^{2} + 8 x + 14$$8$$1$$24$$Q_8$$[2, 3, 4]$
2.8.24.4$x^{8} + 14 x^{4} + 8 x^{3} + 12 x^{2} + 8 x + 14$$8$$1$$24$$Q_8$$[2, 3, 4]$
113Data not computed