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Properties

Label 20.8.10697565742...3125.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{15}\cdot 19^{5}\cdot 1699^{5}$
Root discriminant $44.82$
Ramified primes $5, 19, 1699$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T771

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4639, 8361, 49289, -129410, -56704, 317140, -41661, -139480, -562, 76702, -39538, -8135, 13375, -7056, 1830, -47, -230, 88, -10, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 10*x^18 + 88*x^17 - 230*x^16 - 47*x^15 + 1830*x^14 - 7056*x^13 + 13375*x^12 - 8135*x^11 - 39538*x^10 + 76702*x^9 - 562*x^8 - 139480*x^7 - 41661*x^6 + 317140*x^5 - 56704*x^4 - 129410*x^3 + 49289*x^2 + 8361*x - 4639)
 
gp: K = bnfinit(x^20 - 3*x^19 - 10*x^18 + 88*x^17 - 230*x^16 - 47*x^15 + 1830*x^14 - 7056*x^13 + 13375*x^12 - 8135*x^11 - 39538*x^10 + 76702*x^9 - 562*x^8 - 139480*x^7 - 41661*x^6 + 317140*x^5 - 56704*x^4 - 129410*x^3 + 49289*x^2 + 8361*x - 4639, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 10 x^{18} + 88 x^{17} - 230 x^{16} - 47 x^{15} + 1830 x^{14} - 7056 x^{13} + 13375 x^{12} - 8135 x^{11} - 39538 x^{10} + 76702 x^{9} - 562 x^{8} - 139480 x^{7} - 41661 x^{6} + 317140 x^{5} - 56704 x^{4} - 129410 x^{3} + 49289 x^{2} + 8361 x - 4639 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1069756574259831337445098876953125=5^{15}\cdot 19^{5}\cdot 1699^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.82$
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:  $5, 19, 1699$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{17} a^{18} - \frac{6}{17} a^{17} - \frac{4}{17} a^{16} + \frac{2}{17} a^{15} - \frac{1}{17} a^{14} + \frac{6}{17} a^{12} - \frac{2}{17} a^{11} - \frac{2}{17} a^{10} + \frac{4}{17} a^{9} - \frac{1}{17} a^{8} + \frac{4}{17} a^{7} - \frac{1}{17} a^{6} - \frac{6}{17} a^{5} + \frac{2}{17} a^{4} + \frac{3}{17} a^{3} - \frac{8}{17} a^{2} - \frac{1}{17} a + \frac{3}{17}$, $\frac{1}{951833899532909389068569986419201180532554017131408783127} a^{19} + \frac{19941170261658397969386888922113754199740268746394389742}{951833899532909389068569986419201180532554017131408783127} a^{18} + \frac{258254321555693077502068142842085418642892070439151539161}{951833899532909389068569986419201180532554017131408783127} a^{17} + \frac{230011123905144364530984134800686757192026292879742084853}{951833899532909389068569986419201180532554017131408783127} a^{16} - \frac{417592915625636907286934912205236883079542901784227269135}{951833899532909389068569986419201180532554017131408783127} a^{15} - \frac{223447279566691182223062590346856865406105142114467060313}{951833899532909389068569986419201180532554017131408783127} a^{14} - \frac{276406771326058909437491579077309122139854105787634590370}{951833899532909389068569986419201180532554017131408783127} a^{13} + \frac{45096793299467057231182562968997220662790207029240780308}{951833899532909389068569986419201180532554017131408783127} a^{12} + \frac{48689351153755254822086816209702374013550462726835378385}{951833899532909389068569986419201180532554017131408783127} a^{11} - \frac{392447835700288209403907509463427154825800760296896224284}{951833899532909389068569986419201180532554017131408783127} a^{10} - \frac{187444988775518331502062710969274999761495386001776406629}{951833899532909389068569986419201180532554017131408783127} a^{9} + \frac{342195406230043694736349915196494719721718428939075911464}{951833899532909389068569986419201180532554017131408783127} a^{8} + \frac{119890633138117480703878705916198460563131287421827192225}{951833899532909389068569986419201180532554017131408783127} a^{7} + \frac{319477924006389297076067911432482975730642062977225575948}{951833899532909389068569986419201180532554017131408783127} a^{6} - \frac{206080090049427965234724192547003322900177768123475364279}{951833899532909389068569986419201180532554017131408783127} a^{5} + \frac{404200625710528724612797551160633403614780170643284271046}{951833899532909389068569986419201180532554017131408783127} a^{4} + \frac{223022609286287612456762999336252126569316463513168339223}{951833899532909389068569986419201180532554017131408783127} a^{3} - \frac{142750421028936863518898682701473988080191081007565438631}{951833899532909389068569986419201180532554017131408783127} a^{2} - \frac{14704499151132675855224899771182196395429684811304436377}{55990229384288787592268822730541245913679648066553457831} a - \frac{419110170743860053930737158189496812726472930728211412839}{951833899532909389068569986419201180532554017131408783127}$

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:  $13$
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:  \( 909672090.839 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T771:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n771 are not computed
Character table for t20n771 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.3256446753125.1

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 $20$ $20$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ R $20$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{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$$[\ ]_{2}^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
1699Data not computed