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Properties

Label 20.0.64843561502...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 5^{18}\cdot 11^{10}$
Root discriminant $24.58$
Ramified primes $2, 5, 11$
Class number $2$
Class group $[2]$
Galois group $D_5\wr C_2$ (as 20T50)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, -27, 88, -451, 1252, -2617, 4748, -5627, 5799, -4118, 1550, 72, -463, 381, -72, -83, 102, -63, 26, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 26*x^18 - 63*x^17 + 102*x^16 - 83*x^15 - 72*x^14 + 381*x^13 - 463*x^12 + 72*x^11 + 1550*x^10 - 4118*x^9 + 5799*x^8 - 5627*x^7 + 4748*x^6 - 2617*x^5 + 1252*x^4 - 451*x^3 + 88*x^2 - 27*x + 9)
 
gp: K = bnfinit(x^20 - 7*x^19 + 26*x^18 - 63*x^17 + 102*x^16 - 83*x^15 - 72*x^14 + 381*x^13 - 463*x^12 + 72*x^11 + 1550*x^10 - 4118*x^9 + 5799*x^8 - 5627*x^7 + 4748*x^6 - 2617*x^5 + 1252*x^4 - 451*x^3 + 88*x^2 - 27*x + 9, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 26 x^{18} - 63 x^{17} + 102 x^{16} - 83 x^{15} - 72 x^{14} + 381 x^{13} - 463 x^{12} + 72 x^{11} + 1550 x^{10} - 4118 x^{9} + 5799 x^{8} - 5627 x^{7} + 4748 x^{6} - 2617 x^{5} + 1252 x^{4} - 451 x^{3} + 88 x^{2} - 27 x + 9 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6484356150250000000000000000=2^{16}\cdot 5^{18}\cdot 11^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.58$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{15} - \frac{1}{6} a^{12} + \frac{1}{6} a^{11} - \frac{1}{3} a^{10} - \frac{1}{2} a^{9} + \frac{1}{6} a^{8} - \frac{1}{3} a^{7} + \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{3} + \frac{1}{6} a^{2} - \frac{1}{3} a$, $\frac{1}{30} a^{16} + \frac{1}{15} a^{15} + \frac{1}{10} a^{14} - \frac{2}{15} a^{13} + \frac{1}{6} a^{12} + \frac{1}{5} a^{11} - \frac{7}{30} a^{10} + \frac{1}{30} a^{9} + \frac{2}{5} a^{8} - \frac{1}{2} a^{7} - \frac{1}{30} a^{6} - \frac{1}{15} a^{4} - \frac{3}{10} a^{3} - \frac{1}{2} a^{2} - \frac{1}{30} a - \frac{1}{5}$, $\frac{1}{30} a^{17} - \frac{1}{30} a^{15} + \frac{1}{6} a^{14} - \frac{1}{15} a^{13} - \frac{2}{15} a^{12} - \frac{2}{15} a^{11} - \frac{1}{2} a^{10} - \frac{1}{6} a^{9} + \frac{1}{5} a^{8} - \frac{1}{30} a^{7} - \frac{13}{30} a^{6} + \frac{13}{30} a^{5} - \frac{1}{6} a^{4} - \frac{2}{5} a^{3} - \frac{1}{30} a^{2} + \frac{11}{30} a - \frac{1}{10}$, $\frac{1}{4950} a^{18} - \frac{1}{99} a^{17} + \frac{1}{99} a^{16} + \frac{137}{4950} a^{15} - \frac{119}{825} a^{14} - \frac{128}{825} a^{13} - \frac{101}{550} a^{12} - \frac{48}{275} a^{11} - \frac{877}{4950} a^{10} - \frac{1403}{4950} a^{9} - \frac{274}{825} a^{8} - \frac{2323}{4950} a^{7} + \frac{1091}{2475} a^{6} + \frac{62}{495} a^{5} - \frac{419}{4950} a^{4} - \frac{29}{165} a^{3} - \frac{52}{225} a^{2} - \frac{271}{550} a + \frac{13}{275}$, $\frac{1}{216742613837049754160850} a^{19} + \frac{552313261718377096}{12041256324280541897825} a^{18} - \frac{259574518394143899679}{43348522767409950832170} a^{17} + \frac{248807698868569226192}{36123768972841625693475} a^{16} - \frac{15797979983795142298403}{216742613837049754160850} a^{15} - \frac{894513958768435553461}{14449507589136650277390} a^{14} - \frac{951600201760333079047}{5557502918898711645150} a^{13} + \frac{312535958444226921221}{24082512648561083795650} a^{12} - \frac{25273395197997919914587}{108371306918524877080425} a^{11} + \frac{2114935023094798911286}{8336254378348067467725} a^{10} + \frac{34553454943411108979581}{108371306918524877080425} a^{9} + \frac{10372801564402782849833}{21674261383704975416085} a^{8} + \frac{45834807459364460980283}{216742613837049754160850} a^{7} - \frac{27486446509405521014332}{108371306918524877080425} a^{6} + \frac{22631399124680691676031}{216742613837049754160850} a^{5} + \frac{49346686331481976897534}{108371306918524877080425} a^{4} - \frac{56366205614934028748599}{216742613837049754160850} a^{3} - \frac{26529198212224674373151}{216742613837049754160850} a^{2} + \frac{10183212478981322184373}{24082512648561083795650} a + \frac{2598399136501358467033}{24082512648561083795650}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 784798.952025 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5\wr C_2$ (as 20T50):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 14 conjugacy class representatives for $D_5\wr C_2$
Character table for $D_5\wr C_2$

Intermediate fields

\(\Q(\sqrt{-55}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{-11})\), 10.0.644204000000.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 25 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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.10.13.2$x^{10} + 10 x^{4} + 5$$10$$1$$13$$D_{10}$$[3/2]_{2}^{2}$
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$