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Properties

Label 20.0.38685583611...7424.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 2657^{6}$
Root discriminant $42.60$
Ramified primes $2, 2657$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group 20T1028

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8462, -30108, 88070, -150188, 219800, -217264, 191478, -113416, 68544, -28960, 14200, -4508, 947, -488, 80, -212, 119, 12, -6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 6*x^18 + 12*x^17 + 119*x^16 - 212*x^15 + 80*x^14 - 488*x^13 + 947*x^12 - 4508*x^11 + 14200*x^10 - 28960*x^9 + 68544*x^8 - 113416*x^7 + 191478*x^6 - 217264*x^5 + 219800*x^4 - 150188*x^3 + 88070*x^2 - 30108*x + 8462)
 
gp: K = bnfinit(x^20 - 4*x^19 - 6*x^18 + 12*x^17 + 119*x^16 - 212*x^15 + 80*x^14 - 488*x^13 + 947*x^12 - 4508*x^11 + 14200*x^10 - 28960*x^9 + 68544*x^8 - 113416*x^7 + 191478*x^6 - 217264*x^5 + 219800*x^4 - 150188*x^3 + 88070*x^2 - 30108*x + 8462, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 6 x^{18} + 12 x^{17} + 119 x^{16} - 212 x^{15} + 80 x^{14} - 488 x^{13} + 947 x^{12} - 4508 x^{11} + 14200 x^{10} - 28960 x^{9} + 68544 x^{8} - 113416 x^{7} + 191478 x^{6} - 217264 x^{5} + 219800 x^{4} - 150188 x^{3} + 88070 x^{2} - 30108 x + 8462 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(386855836117226742051972447207424=2^{40}\cdot 2657^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.60$
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:  $2, 2657$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{91894586667404178522673852282649713187904079681} a^{19} - \frac{33817681668980937020917259111960985881877089437}{91894586667404178522673852282649713187904079681} a^{18} + \frac{44617367650340655788382359204852949742318125084}{91894586667404178522673852282649713187904079681} a^{17} + \frac{974425951564338752867777903616151725973769043}{4836557193021272553824939593823669115152846299} a^{16} - \frac{44113699258455339433711628321564401419800525301}{91894586667404178522673852282649713187904079681} a^{15} - \frac{23769733883711559861534892738015279595143185446}{91894586667404178522673852282649713187904079681} a^{14} + \frac{27596000674508591049694948683930991535429862962}{91894586667404178522673852282649713187904079681} a^{13} + \frac{25841091550541238392837871871715937817700926768}{91894586667404178522673852282649713187904079681} a^{12} + \frac{13213824198885780214406355791263979421381107171}{91894586667404178522673852282649713187904079681} a^{11} + \frac{34497367620659901506974849031050386896375398897}{91894586667404178522673852282649713187904079681} a^{10} - \frac{1510363448210525229545097084344375116886267457}{4836557193021272553824939593823669115152846299} a^{9} - \frac{25282663171344599336275353537428923178015620648}{91894586667404178522673852282649713187904079681} a^{8} - \frac{2166569230091817148143811081062968316220674362}{4836557193021272553824939593823669115152846299} a^{7} + \frac{41123712128040301220589942553130753209199218589}{91894586667404178522673852282649713187904079681} a^{6} + \frac{322132093989025561075669392396716412010837837}{2483637477497410230342536548179721978051461613} a^{5} - \frac{666206565833791543298658130439246416893381328}{2483637477497410230342536548179721978051461613} a^{4} - \frac{18989280656336201812223830414379495057089480492}{91894586667404178522673852282649713187904079681} a^{3} + \frac{2605839787741042232167014177103856016799609846}{91894586667404178522673852282649713187904079681} a^{2} + \frac{42948819849295841738875243930521911906798619167}{91894586667404178522673852282649713187904079681} a + \frac{5243668411408393640496211753568180130195939783}{91894586667404178522673852282649713187904079681}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  $9$
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:  \( 27669372.8595 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1028:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 228 conjugacy class representatives for t20n1028 are not computed
Character table for t20n1028 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.6.925322313728.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$2.8.16.13$x^{8} + 6 x^{6} + 4 x^{5} + 2 x^{4} + 4$$4$$2$$16$$D_4\times C_2$$[2, 2, 3]^{2}$
2.12.24.342$x^{12} + 4 x^{11} + 2 x^{10} + 4 x^{9} + 4 x^{5} - 2 x^{4} + 4 x^{3} - 2 x^{2} + 4 x - 2$$12$$1$$24$$C_2 \times S_4$$[4/3, 4/3, 3]_{3}^{2}$
2657Data not computed