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Properties

Label 20.6.18079489015...9712.6
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{10}\cdot 11^{16}\cdot 727^{3}$
Root discriminant $25.87$
Ramified primes $2, 11, 727$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T747

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![67, -485, -1640, -482, 1712, 178, -1980, -66, 1625, -469, -862, 751, 71, -295, 84, 31, -39, 16, 4, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 4*x^18 + 16*x^17 - 39*x^16 + 31*x^15 + 84*x^14 - 295*x^13 + 71*x^12 + 751*x^11 - 862*x^10 - 469*x^9 + 1625*x^8 - 66*x^7 - 1980*x^6 + 178*x^5 + 1712*x^4 - 482*x^3 - 1640*x^2 - 485*x + 67)
 
gp: K = bnfinit(x^20 - 5*x^19 + 4*x^18 + 16*x^17 - 39*x^16 + 31*x^15 + 84*x^14 - 295*x^13 + 71*x^12 + 751*x^11 - 862*x^10 - 469*x^9 + 1625*x^8 - 66*x^7 - 1980*x^6 + 178*x^5 + 1712*x^4 - 482*x^3 - 1640*x^2 - 485*x + 67, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 4 x^{18} + 16 x^{17} - 39 x^{16} + 31 x^{15} + 84 x^{14} - 295 x^{13} + 71 x^{12} + 751 x^{11} - 862 x^{10} - 469 x^{9} + 1625 x^{8} - 66 x^{7} - 1980 x^{6} + 178 x^{5} + 1712 x^{4} - 482 x^{3} - 1640 x^{2} - 485 x + 67 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-18079489015266793067734899712=-\,2^{10}\cdot 11^{16}\cdot 727^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.87$
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:  $2, 11, 727$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{23} a^{18} - \frac{1}{23} a^{17} + \frac{11}{23} a^{16} + \frac{3}{23} a^{15} + \frac{2}{23} a^{14} + \frac{3}{23} a^{13} + \frac{3}{23} a^{12} + \frac{3}{23} a^{11} + \frac{1}{23} a^{10} + \frac{6}{23} a^{9} + \frac{1}{23} a^{8} - \frac{8}{23} a^{7} - \frac{6}{23} a^{6} + \frac{6}{23} a^{5} + \frac{2}{23} a^{4} - \frac{1}{23} a^{3} + \frac{5}{23} a^{2} + \frac{10}{23} a - \frac{4}{23}$, $\frac{1}{2999165897484223479772769082677} a^{19} + \frac{9861715436344850013912569547}{2999165897484223479772769082677} a^{18} + \frac{772018158722790969139434023479}{2999165897484223479772769082677} a^{17} - \frac{475028992524612666392857610423}{2999165897484223479772769082677} a^{16} + \frac{1469377472428742071939300120236}{2999165897484223479772769082677} a^{15} + \frac{347193872933285695456017515917}{2999165897484223479772769082677} a^{14} + \frac{1181761537211794908242305510567}{2999165897484223479772769082677} a^{13} + \frac{831917447513517430994868185662}{2999165897484223479772769082677} a^{12} + \frac{221191441382363486949776657677}{2999165897484223479772769082677} a^{11} - \frac{959947899834478949931861437648}{2999165897484223479772769082677} a^{10} + \frac{1365801875675047998037790974348}{2999165897484223479772769082677} a^{9} - \frac{910998644265226160731630724079}{2999165897484223479772769082677} a^{8} + \frac{397951458792876931816716354902}{2999165897484223479772769082677} a^{7} - \frac{1097139963411660263952265154774}{2999165897484223479772769082677} a^{6} - \frac{778628927048187380024877516066}{2999165897484223479772769082677} a^{5} + \frac{56402208538269027265622967845}{130398517281922759990120394899} a^{4} + \frac{2185970797090305459681230669}{15071185414493585325491301923} a^{3} + \frac{874038355401905442054888900627}{2999165897484223479772769082677} a^{2} - \frac{1401766570868097882982316030293}{2999165897484223479772769082677} a - \frac{327436550833028578779662544872}{2999165897484223479772769082677}$

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:  $12$
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:  \( 2087158.6212 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T747:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n747 are not computed
Character table for t20n747 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.8.155838906487.1

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

Sibling fields

Degree 20 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 $20$ $20$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $20$

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$2.10.10.6$x^{10} - 5 x^{8} - 18 x^{6} - 46 x^{4} + 49 x^{2} - 13$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
11Data not computed
727Data not computed