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Properties

Label 20.10.2955425558...4411.2
Degree $20$
Signature $[10, 5]$
Discriminant $-\,11^{16}\cdot 1451^{5}$
Root discriminant $42.03$
Ramified primes $11, 1451$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T310

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![43, -112, -3449, 2101, 15773, -33332, 37707, -42349, 48560, -40810, 20428, -3559, -3077, 3464, -1921, 546, 8, -67, 30, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 30*x^18 - 67*x^17 + 8*x^16 + 546*x^15 - 1921*x^14 + 3464*x^13 - 3077*x^12 - 3559*x^11 + 20428*x^10 - 40810*x^9 + 48560*x^8 - 42349*x^7 + 37707*x^6 - 33332*x^5 + 15773*x^4 + 2101*x^3 - 3449*x^2 - 112*x + 43)
 
gp: K = bnfinit(x^20 - 8*x^19 + 30*x^18 - 67*x^17 + 8*x^16 + 546*x^15 - 1921*x^14 + 3464*x^13 - 3077*x^12 - 3559*x^11 + 20428*x^10 - 40810*x^9 + 48560*x^8 - 42349*x^7 + 37707*x^6 - 33332*x^5 + 15773*x^4 + 2101*x^3 - 3449*x^2 - 112*x + 43, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 30 x^{18} - 67 x^{17} + 8 x^{16} + 546 x^{15} - 1921 x^{14} + 3464 x^{13} - 3077 x^{12} - 3559 x^{11} + 20428 x^{10} - 40810 x^{9} + 48560 x^{8} - 42349 x^{7} + 37707 x^{6} - 33332 x^{5} + 15773 x^{4} + 2101 x^{3} - 3449 x^{2} - 112 x + 43 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-295542555810828925413205326764411=-\,11^{16}\cdot 1451^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.03$
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:  $11, 1451$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11} a^{15} + \frac{5}{11} a^{14} + \frac{1}{11} a^{13} + \frac{2}{11} a^{12} + \frac{3}{11} a^{11} - \frac{5}{11} a^{9} - \frac{4}{11} a^{8} + \frac{3}{11} a^{6} + \frac{2}{11} a^{5} - \frac{4}{11} a^{4} - \frac{5}{11} a^{3} - \frac{1}{11} a^{2} + \frac{2}{11} a - \frac{1}{11}$, $\frac{1}{11} a^{16} - \frac{2}{11} a^{14} - \frac{3}{11} a^{13} + \frac{4}{11} a^{12} - \frac{4}{11} a^{11} - \frac{5}{11} a^{10} - \frac{1}{11} a^{9} - \frac{2}{11} a^{8} + \frac{3}{11} a^{7} - \frac{2}{11} a^{6} - \frac{3}{11} a^{5} + \frac{4}{11} a^{4} + \frac{2}{11} a^{3} - \frac{4}{11} a^{2} + \frac{5}{11}$, $\frac{1}{11} a^{17} - \frac{4}{11} a^{14} - \frac{5}{11} a^{13} + \frac{1}{11} a^{11} - \frac{1}{11} a^{10} - \frac{1}{11} a^{9} - \frac{5}{11} a^{8} - \frac{2}{11} a^{7} + \frac{3}{11} a^{6} - \frac{3}{11} a^{5} + \frac{5}{11} a^{4} - \frac{3}{11} a^{3} - \frac{2}{11} a^{2} - \frac{2}{11} a - \frac{2}{11}$, $\frac{1}{11} a^{18} + \frac{4}{11} a^{14} + \frac{4}{11} a^{13} - \frac{2}{11} a^{12} - \frac{1}{11} a^{10} - \frac{3}{11} a^{9} + \frac{4}{11} a^{8} + \frac{3}{11} a^{7} - \frac{2}{11} a^{6} + \frac{2}{11} a^{5} + \frac{3}{11} a^{4} + \frac{5}{11} a^{2} - \frac{5}{11} a - \frac{4}{11}$, $\frac{1}{103394365798326220045338134139289} a^{19} - \frac{4354461631178952142477474545931}{103394365798326220045338134139289} a^{18} + \frac{272281957433742727560719782715}{103394365798326220045338134139289} a^{17} + \frac{4476250353613482696861355787912}{103394365798326220045338134139289} a^{16} - \frac{832285855241272461117267220264}{103394365798326220045338134139289} a^{15} + \frac{41500678151681629502340696086885}{103394365798326220045338134139289} a^{14} - \frac{16868682688205206242784702541348}{103394365798326220045338134139289} a^{13} - \frac{60860117784412847863560569889}{408673382602079921127818712013} a^{12} - \frac{21882242715500873150350155383561}{103394365798326220045338134139289} a^{11} + \frac{32224346097706018012803780261960}{103394365798326220045338134139289} a^{10} + \frac{32496162814872674888205362085732}{103394365798326220045338134139289} a^{9} - \frac{2492284092025276514964792163496}{9399487799847838185939830376299} a^{8} + \frac{3285473831791496147976488749785}{9399487799847838185939830376299} a^{7} + \frac{41348957452489925643648898059234}{103394365798326220045338134139289} a^{6} - \frac{13906236399765211968922224963926}{103394365798326220045338134139289} a^{5} + \frac{28440044332053372491544903554792}{103394365798326220045338134139289} a^{4} + \frac{41486455412892495958513355021135}{103394365798326220045338134139289} a^{3} + \frac{19772840069670659915551174307051}{103394365798326220045338134139289} a^{2} - \frac{33546607007344074592186548047040}{103394365798326220045338134139289} a - \frac{1522094053115394623109636287922}{9399487799847838185939830376299}$

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:  $14$
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:  \( 810275651.403 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T310:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 44 conjugacy class representatives for t20n310
Character table for t20n310 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.451311402416281.2

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
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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
1451Data not computed