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Properties

Label 20.4.26031679515...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{16}\cdot 5^{22}\cdot 7^{8}\cdot 17^{2}$
Root discriminant $29.56$
Ramified primes $2, 5, 7, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T140

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![239, -640, -1480, -2830, 1495, 9546, 8210, 4930, 5500, 3730, 1470, 390, -200, -530, -210, -122, 5, -10, 10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 10*x^18 - 10*x^17 + 5*x^16 - 122*x^15 - 210*x^14 - 530*x^13 - 200*x^12 + 390*x^11 + 1470*x^10 + 3730*x^9 + 5500*x^8 + 4930*x^7 + 8210*x^6 + 9546*x^5 + 1495*x^4 - 2830*x^3 - 1480*x^2 - 640*x + 239)
 
gp: K = bnfinit(x^20 + 10*x^18 - 10*x^17 + 5*x^16 - 122*x^15 - 210*x^14 - 530*x^13 - 200*x^12 + 390*x^11 + 1470*x^10 + 3730*x^9 + 5500*x^8 + 4930*x^7 + 8210*x^6 + 9546*x^5 + 1495*x^4 - 2830*x^3 - 1480*x^2 - 640*x + 239, 1)
 

Normalized defining polynomial

\( x^{20} + 10 x^{18} - 10 x^{17} + 5 x^{16} - 122 x^{15} - 210 x^{14} - 530 x^{13} - 200 x^{12} + 390 x^{11} + 1470 x^{10} + 3730 x^{9} + 5500 x^{8} + 4930 x^{7} + 8210 x^{6} + 9546 x^{5} + 1495 x^{4} - 2830 x^{3} - 1480 x^{2} - 640 x + 239 \)

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:  \(260316795156250000000000000000=2^{16}\cdot 5^{22}\cdot 7^{8}\cdot 17^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.56$
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:  $2, 5, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{4} a^{4} + \frac{1}{4}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{5} - \frac{3}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{3}{8} a + \frac{1}{8}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{3}{8} a^{4} + \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{3}{8}$, $\frac{1}{16} a^{15} - \frac{1}{16} a^{14} - \frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{3}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{16} a^{7} - \frac{3}{16} a^{6} - \frac{3}{16} a^{5} + \frac{1}{16} a^{4} - \frac{7}{16} a^{3} - \frac{3}{16} a^{2} - \frac{1}{16} a - \frac{7}{16}$, $\frac{1}{16} a^{16} - \frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} + \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{8} a^{3} - \frac{3}{8} a^{2} - \frac{1}{8} a + \frac{5}{16}$, $\frac{1}{16} a^{17} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} + \frac{1}{8} a^{3} + \frac{3}{8} a^{2} - \frac{1}{16} a - \frac{1}{8}$, $\frac{1}{32} a^{18} - \frac{1}{32} a^{16} - \frac{1}{16} a^{14} - \frac{1}{16} a^{13} - \frac{1}{8} a^{10} + \frac{3}{16} a^{9} + \frac{3}{16} a^{8} - \frac{1}{4} a^{7} - \frac{3}{16} a^{6} - \frac{3}{16} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{5}{32} a^{2} - \frac{7}{16} a - \frac{5}{32}$, $\frac{1}{603508364640013979303410100969568} a^{19} + \frac{1132418997047338237442074799555}{150877091160003494825852525242392} a^{18} + \frac{16955835866435033658718766967521}{603508364640013979303410100969568} a^{17} - \frac{162274502857645561399328468987}{6286545465000145617743855218433} a^{16} - \frac{3818621837622914121666204508643}{301754182320006989651705050484784} a^{15} - \frac{9077446568212393315197819679571}{301754182320006989651705050484784} a^{14} + \frac{5603345503612210842346147225453}{150877091160003494825852525242392} a^{13} - \frac{501903372984781023753192544821}{50292363720001164941950841747464} a^{12} - \frac{4119058058394517468390777783535}{150877091160003494825852525242392} a^{11} - \frac{27793464516801284858341544946761}{301754182320006989651705050484784} a^{10} + \frac{35520897684214345502268583030277}{301754182320006989651705050484784} a^{9} - \frac{367918100732938569089576546553}{6286545465000145617743855218433} a^{8} - \frac{52275751264019265547885864379737}{301754182320006989651705050484784} a^{7} + \frac{13750504458253582471452873508617}{100584727440002329883901683494928} a^{6} - \frac{9555002536338525050060518046845}{150877091160003494825852525242392} a^{5} - \frac{57283799609722670991594584347409}{150877091160003494825852525242392} a^{4} - \frac{4506261564548230833476837072895}{201169454880004659767803366989856} a^{3} + \frac{63692111902950173224537882731283}{301754182320006989651705050484784} a^{2} + \frac{68346479134130361097608608550903}{201169454880004659767803366989856} a - \frac{10586497910331413255053014300361}{37719272790000873706463131310598}$

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:  \( 10386308.2947 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T140:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 22 conjugacy class representatives for t20n140
Character table for t20n140 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.2450000.1, 10.2.102042500000000.1, 10.2.510212500000000.1, 10.10.30012500000000.1

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 32 siblings: 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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}$
5Data not computed
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
17.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$