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Properties

Label 20.12.5834302965...2368.3
Degree $20$
Signature $[12, 4]$
Discriminant $2^{10}\cdot 19^{10}\cdot 43^{11}$
Root discriminant $48.79$
Ramified primes $2, 19, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T423

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![311, -37560, -145395, -129423, 168164, 399520, 254736, 99646, 67699, -11581, -23958, -622, -4342, -586, 1228, -227, 5, 69, -17, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 17*x^18 + 69*x^17 + 5*x^16 - 227*x^15 + 1228*x^14 - 586*x^13 - 4342*x^12 - 622*x^11 - 23958*x^10 - 11581*x^9 + 67699*x^8 + 99646*x^7 + 254736*x^6 + 399520*x^5 + 168164*x^4 - 129423*x^3 - 145395*x^2 - 37560*x + 311)
 
gp: K = bnfinit(x^20 - 4*x^19 - 17*x^18 + 69*x^17 + 5*x^16 - 227*x^15 + 1228*x^14 - 586*x^13 - 4342*x^12 - 622*x^11 - 23958*x^10 - 11581*x^9 + 67699*x^8 + 99646*x^7 + 254736*x^6 + 399520*x^5 + 168164*x^4 - 129423*x^3 - 145395*x^2 - 37560*x + 311, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 17 x^{18} + 69 x^{17} + 5 x^{16} - 227 x^{15} + 1228 x^{14} - 586 x^{13} - 4342 x^{12} - 622 x^{11} - 23958 x^{10} - 11581 x^{9} + 67699 x^{8} + 99646 x^{7} + 254736 x^{6} + 399520 x^{5} + 168164 x^{4} - 129423 x^{3} - 145395 x^{2} - 37560 x + 311 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5834302965409512291058019417402368=2^{10}\cdot 19^{10}\cdot 43^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.79$
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:  $2, 19, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{16} + \frac{1}{4} a^{15} + \frac{1}{4} a^{13} + \frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{1240} a^{18} - \frac{9}{155} a^{17} - \frac{47}{310} a^{16} - \frac{523}{1240} a^{15} - \frac{23}{248} a^{14} + \frac{157}{620} a^{13} - \frac{379}{1240} a^{12} + \frac{67}{310} a^{11} - \frac{613}{1240} a^{10} - \frac{7}{620} a^{9} + \frac{85}{248} a^{8} - \frac{543}{1240} a^{7} - \frac{3}{10} a^{6} - \frac{477}{1240} a^{5} - \frac{48}{155} a^{4} + \frac{511}{1240} a^{3} - \frac{42}{155} a^{2} - \frac{13}{310} a - \frac{507}{1240}$, $\frac{1}{105749458188328912068681515700549353314684228625200} a^{19} - \frac{5998976767009866808437577724295509933592909261}{21149891637665782413736303140109870662936845725040} a^{18} + \frac{25940597237910424776699734609377549174314978783}{213204552799050225944922410686591438134444009325} a^{17} + \frac{9991339809796420546751139856695889993129373134141}{105749458188328912068681515700549353314684228625200} a^{16} - \frac{941856837856506179029367562837866301174589165419}{26437364547082228017170378925137338328671057156300} a^{15} + \frac{10670437793437306342018424780437634889600582892309}{105749458188328912068681515700549353314684228625200} a^{14} + \frac{25261880535978466013651118327050180405627153891119}{105749458188328912068681515700549353314684228625200} a^{13} - \frac{4225324913009894889895996053621638891775620974537}{21149891637665782413736303140109870662936845725040} a^{12} - \frac{2627812379665840847510077446456249872765634815377}{105749458188328912068681515700549353314684228625200} a^{11} + \frac{2218288791731251045684117077553545088302276056739}{21149891637665782413736303140109870662936845725040} a^{10} + \frac{1484355024615960485678224680589294343058588055597}{3411272844784803615118758570985463010151104149200} a^{9} - \frac{1359813748229441824864289406921221149916650168377}{26437364547082228017170378925137338328671057156300} a^{8} + \frac{37644532915101355070080224962211956449929592953107}{105749458188328912068681515700549353314684228625200} a^{7} - \frac{2829652597165409264861224082498245225762035719701}{105749458188328912068681515700549353314684228625200} a^{6} - \frac{49511874406592333714521293585033105524357477336423}{105749458188328912068681515700549353314684228625200} a^{5} - \frac{37284185026257572795899836917125083892765816457617}{105749458188328912068681515700549353314684228625200} a^{4} + \frac{15156000398920154937313922546172391833323009888961}{105749458188328912068681515700549353314684228625200} a^{3} + \frac{30425475148539425196160194160951391701988101097}{90538919681788452113597188099785405235174853275} a^{2} + \frac{606484308589248181209610536761965953074965838651}{1338600736561125469223816654437333586261825678800} a + \frac{3709434667567695431407390162203634272636804097551}{105749458188328912068681515700549353314684228625200}$

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:  $15$
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:  \( 5938239545.27 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T423:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 160 conjugacy class representatives for t20n423 are not computed
Character table for t20n423 is not computed

Intermediate fields

\(\Q(\sqrt{817}) \), 5.5.667489.1 x5, 10.10.364007458703857.1

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
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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.12$x^{10} - 11 x^{8} + 54 x^{6} - 10 x^{4} + 9 x^{2} - 11$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$43$43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.4.3.1$x^{4} + 387$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$