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Properties

Label 20.20.2567793171...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{16}\cdot 5^{30}\cdot 29^{10}$
Root discriminant $104.83$
Ramified primes $2, 5, 29$
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![-6385664, 25927680, -14233600, -49489920, 48076800, 32580096, -42205440, -9676800, 17496960, 1478400, -4098224, -120960, 582400, 5040, -51200, -84, 2720, 0, -80, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 80*x^18 + 2720*x^16 - 84*x^15 - 51200*x^14 + 5040*x^13 + 582400*x^12 - 120960*x^11 - 4098224*x^10 + 1478400*x^9 + 17496960*x^8 - 9676800*x^7 - 42205440*x^6 + 32580096*x^5 + 48076800*x^4 - 49489920*x^3 - 14233600*x^2 + 25927680*x - 6385664)
 
gp: K = bnfinit(x^20 - 80*x^18 + 2720*x^16 - 84*x^15 - 51200*x^14 + 5040*x^13 + 582400*x^12 - 120960*x^11 - 4098224*x^10 + 1478400*x^9 + 17496960*x^8 - 9676800*x^7 - 42205440*x^6 + 32580096*x^5 + 48076800*x^4 - 49489920*x^3 - 14233600*x^2 + 25927680*x - 6385664, 1)
 

Normalized defining polynomial

\( x^{20} - 80 x^{18} + 2720 x^{16} - 84 x^{15} - 51200 x^{14} + 5040 x^{13} + 582400 x^{12} - 120960 x^{11} - 4098224 x^{10} + 1478400 x^{9} + 17496960 x^{8} - 9676800 x^{7} - 42205440 x^{6} + 32580096 x^{5} + 48076800 x^{4} - 49489920 x^{3} - 14233600 x^{2} + 25927680 x - 6385664 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(25677931719982971191406250000000000000000=2^{16}\cdot 5^{30}\cdot 29^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $104.83$
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:  $2, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{4} a^{6}$, $\frac{1}{4} a^{7}$, $\frac{1}{8} a^{8}$, $\frac{1}{8} a^{9}$, $\frac{1}{240} a^{10} - \frac{1}{24} a^{8} + \frac{1}{12} a^{6} - \frac{1}{20} a^{5} + \frac{1}{6} a^{4} - \frac{1}{3} a^{2} + \frac{1}{15}$, $\frac{1}{240} a^{11} - \frac{1}{24} a^{9} + \frac{1}{12} a^{7} - \frac{1}{20} a^{6} - \frac{1}{12} a^{5} + \frac{1}{6} a^{3} + \frac{1}{15} a$, $\frac{1}{480} a^{12} - \frac{1}{24} a^{8} - \frac{1}{40} a^{7} + \frac{1}{6} a^{4} + \frac{11}{30} a^{2} + \frac{1}{3}$, $\frac{1}{960} a^{13} + \frac{1}{24} a^{9} - \frac{1}{80} a^{8} + \frac{1}{12} a^{5} + \frac{11}{60} a^{3} - \frac{1}{3} a$, $\frac{1}{9600} a^{14} - \frac{1}{2400} a^{13} - \frac{1}{2400} a^{12} + \frac{1}{600} a^{11} + \frac{1}{600} a^{10} + \frac{127}{2400} a^{9} + \frac{23}{600} a^{8} - \frac{13}{150} a^{7} + \frac{1}{75} a^{6} + \frac{1}{75} a^{5} + \frac{27}{200} a^{4} - \frac{31}{150} a^{3} - \frac{31}{150} a^{2} + \frac{4}{25} a + \frac{37}{75}$, $\frac{1}{76800} a^{15} + \frac{1}{3840} a^{13} - \frac{1}{480} a^{11} + \frac{23}{19200} a^{10} + \frac{1}{48} a^{9} - \frac{29}{480} a^{8} + \frac{1}{12} a^{7} - \frac{19}{240} a^{6} + \frac{73}{4800} a^{5} + \frac{1}{6} a^{4} + \frac{17}{80} a^{3} - \frac{1}{3} a^{2} - \frac{9}{20} a - \frac{31}{300}$, $\frac{1}{76800} a^{16} - \frac{1}{19200} a^{14} + \frac{1}{4800} a^{13} - \frac{1}{1200} a^{12} + \frac{7}{19200} a^{11} - \frac{1}{1200} a^{10} - \frac{21}{400} a^{9} + \frac{9}{400} a^{8} + \frac{17}{1200} a^{7} + \frac{147}{1600} a^{6} - \frac{9}{100} a^{5} + \frac{169}{1200} a^{4} - \frac{23}{100} a^{3} - \frac{149}{300} a^{2} - \frac{11}{60} a + \frac{19}{75}$, $\frac{1}{768000} a^{17} - \frac{1}{384000} a^{16} - \frac{1}{192000} a^{15} - \frac{1}{19200} a^{14} + \frac{1}{2400} a^{13} + \frac{103}{192000} a^{12} + \frac{97}{96000} a^{11} + \frac{23}{12000} a^{10} - \frac{37}{800} a^{9} + \frac{13}{2400} a^{8} - \frac{1589}{16000} a^{7} - \frac{611}{8000} a^{6} - \frac{143}{12000} a^{5} - \frac{211}{1200} a^{4} + \frac{59}{600} a^{3} + \frac{1139}{3000} a^{2} + \frac{611}{1500} a - \frac{14}{125}$, $\frac{1}{1536000} a^{18} - \frac{1}{192000} a^{16} - \frac{1}{192000} a^{15} - \frac{1}{19200} a^{14} - \frac{17}{384000} a^{13} - \frac{1}{4800} a^{12} + \frac{23}{32000} a^{11} - \frac{21}{16000} a^{10} - \frac{17}{4800} a^{9} - \frac{269}{32000} a^{8} + \frac{41}{400} a^{7} - \frac{19}{375} a^{6} - \frac{51}{4000} a^{5} - \frac{11}{75} a^{4} - \frac{1441}{6000} a^{3} + \frac{89}{300} a^{2} - \frac{51}{500} a - \frac{229}{750}$, $\frac{1}{3072000} a^{19} + \frac{1}{192000} a^{16} + \frac{1}{384000} a^{15} + \frac{23}{768000} a^{14} - \frac{1}{6400} a^{13} - \frac{3}{12800} a^{12} - \frac{19}{16000} a^{11} - \frac{17}{96000} a^{10} + \frac{7673}{192000} a^{9} - \frac{59}{1200} a^{8} - \frac{13}{1600} a^{7} - \frac{279}{4000} a^{6} - \frac{2357}{24000} a^{5} - \frac{577}{4000} a^{4} + \frac{187}{1200} a^{3} - \frac{7}{200} a^{2} - \frac{109}{500} a - \frac{641}{1500}$

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:  $19$
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:  \( 94753821362700 \) (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

\(\Q(\sqrt{145}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{29})\), 10.10.32048670312500000000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}$ R ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$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$5.10.13.2$x^{10} + 10 x^{4} + 5$$10$$1$$13$$D_{10}$$[3/2]_{2}^{2}$
5.10.17.6$x^{10} - 20 x^{8} + 5$$10$$1$$17$$D_{10}$$[2]_{2}^{2}$
$29$29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$