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Properties

Label 20.0.24156109058...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{30}\cdot 11^{10}$
Root discriminant $37.08$
Ramified primes $5, 11$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
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![199, -70, 1175, -1070, 2975, -3014, 4145, -3600, 4720, -2200, 4047, -720, 2275, -120, 800, -8, 170, 0, 20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 20*x^18 + 170*x^16 - 8*x^15 + 800*x^14 - 120*x^13 + 2275*x^12 - 720*x^11 + 4047*x^10 - 2200*x^9 + 4720*x^8 - 3600*x^7 + 4145*x^6 - 3014*x^5 + 2975*x^4 - 1070*x^3 + 1175*x^2 - 70*x + 199)
 
gp: K = bnfinit(x^20 + 20*x^18 + 170*x^16 - 8*x^15 + 800*x^14 - 120*x^13 + 2275*x^12 - 720*x^11 + 4047*x^10 - 2200*x^9 + 4720*x^8 - 3600*x^7 + 4145*x^6 - 3014*x^5 + 2975*x^4 - 1070*x^3 + 1175*x^2 - 70*x + 199, 1)
 

Normalized defining polynomial

\( x^{20} + 20 x^{18} + 170 x^{16} - 8 x^{15} + 800 x^{14} - 120 x^{13} + 2275 x^{12} - 720 x^{11} + 4047 x^{10} - 2200 x^{9} + 4720 x^{8} - 3600 x^{7} + 4145 x^{6} - 3014 x^{5} + 2975 x^{4} - 1070 x^{3} + 1175 x^{2} - 70 x + 199 \)

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:  \(24156109058298170566558837890625=5^{30}\cdot 11^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.08$
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:  $5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{3} a^{5} - \frac{1}{6} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{1}{3} a^{6} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{1}{3} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{6} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} + \frac{1}{3} a^{8} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{930} a^{15} + \frac{1}{62} a^{13} - \frac{13}{186} a^{11} + \frac{113}{930} a^{10} + \frac{55}{186} a^{9} + \frac{20}{93} a^{8} - \frac{65}{186} a^{7} + \frac{13}{31} a^{6} + \frac{10}{31} a^{5} + \frac{15}{62} a^{4} + \frac{43}{186} a^{3} + \frac{19}{93} a^{2} + \frac{40}{93} a - \frac{127}{465}$, $\frac{1}{930} a^{16} + \frac{1}{62} a^{14} - \frac{13}{186} a^{12} - \frac{7}{155} a^{11} - \frac{19}{93} a^{10} + \frac{20}{93} a^{9} - \frac{65}{186} a^{8} - \frac{77}{186} a^{7} - \frac{1}{93} a^{6} + \frac{38}{93} a^{5} + \frac{37}{93} a^{4} - \frac{43}{93} a^{3} + \frac{3}{31} a^{2} + \frac{61}{155} a + \frac{1}{6}$, $\frac{1}{930} a^{17} + \frac{2}{93} a^{13} - \frac{7}{155} a^{12} + \frac{1}{93} a^{11} - \frac{10}{93} a^{10} - \frac{11}{93} a^{9} + \frac{5}{186} a^{8} - \frac{25}{93} a^{7} + \frac{11}{93} a^{6} - \frac{17}{62} a^{5} + \frac{38}{93} a^{4} - \frac{7}{186} a^{3} + \frac{51}{155} a^{2} + \frac{3}{62} a - \frac{13}{186}$, $\frac{1}{4650} a^{18} + \frac{1}{2325} a^{17} - \frac{1}{2325} a^{16} - \frac{1}{4650} a^{15} - \frac{11}{310} a^{14} + \frac{23}{775} a^{13} + \frac{61}{775} a^{12} + \frac{112}{2325} a^{11} - \frac{353}{4650} a^{10} + \frac{64}{155} a^{9} + \frac{151}{310} a^{8} + \frac{109}{310} a^{7} + \frac{83}{186} a^{6} + \frac{103}{930} a^{5} + \frac{71}{186} a^{4} - \frac{34}{775} a^{3} - \frac{111}{1550} a^{2} - \frac{369}{1550} a - \frac{71}{150}$, $\frac{1}{4650} a^{19} - \frac{1}{4650} a^{17} - \frac{1}{2325} a^{16} + \frac{1}{2325} a^{15} - \frac{191}{2325} a^{14} + \frac{34}{465} a^{13} + \frac{191}{2325} a^{12} + \frac{359}{4650} a^{11} - \frac{377}{2325} a^{10} - \frac{23}{93} a^{9} - \frac{139}{930} a^{8} - \frac{67}{465} a^{7} + \frac{164}{465} a^{6} - \frac{117}{310} a^{5} - \frac{743}{1550} a^{4} - \frac{40}{93} a^{3} + \frac{989}{4650} a^{2} - \frac{489}{1550} a - \frac{704}{2325}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 19342033.3142 \) (assuming GRH)
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

\(\Q(\sqrt{-55}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{-11})\), 10.6.3692626953125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.10.15.8$x^{10} - 10 x^{6} - 10 x^{5} + 5$$10$$1$$15$$C_5^2 : C_4$$[5/4, 7/4]_{4}$
5.10.15.8$x^{10} - 10 x^{6} - 10 x^{5} + 5$$10$$1$$15$$C_5^2 : C_4$$[5/4, 7/4]_{4}$
$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}$