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Properties

Label 20.8.31436627350...3241.1
Degree $20$
Signature $[8, 6]$
Discriminant $3^{4}\cdot 19^{2}\cdot 401^{10}$
Root discriminant $33.49$
Ramified primes $3, 19, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T347

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9, 135, -384, -702, 986, 1302, 289, 168, -433, -820, -665, -790, -553, -137, 77, 128, 61, -9, -14, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 14*x^18 - 9*x^17 + 61*x^16 + 128*x^15 + 77*x^14 - 137*x^13 - 553*x^12 - 790*x^11 - 665*x^10 - 820*x^9 - 433*x^8 + 168*x^7 + 289*x^6 + 1302*x^5 + 986*x^4 - 702*x^3 - 384*x^2 + 135*x - 9)
 
gp: K = bnfinit(x^20 - 14*x^18 - 9*x^17 + 61*x^16 + 128*x^15 + 77*x^14 - 137*x^13 - 553*x^12 - 790*x^11 - 665*x^10 - 820*x^9 - 433*x^8 + 168*x^7 + 289*x^6 + 1302*x^5 + 986*x^4 - 702*x^3 - 384*x^2 + 135*x - 9, 1)
 

Normalized defining polynomial

\( x^{20} - 14 x^{18} - 9 x^{17} + 61 x^{16} + 128 x^{15} + 77 x^{14} - 137 x^{13} - 553 x^{12} - 790 x^{11} - 665 x^{10} - 820 x^{9} - 433 x^{8} + 168 x^{7} + 289 x^{6} + 1302 x^{5} + 986 x^{4} - 702 x^{3} - 384 x^{2} + 135 x - 9 \)

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:  \(3143662735061279800077532193241=3^{4}\cdot 19^{2}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.49$
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:  $3, 19, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{14} - \frac{1}{9} a^{13} + \frac{4}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{2}{9} a^{8} - \frac{4}{9} a^{7} - \frac{2}{9} a^{6} + \frac{4}{9} a^{5} + \frac{2}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{15} - \frac{1}{9} a^{14} + \frac{4}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{2}{9} a^{9} - \frac{4}{9} a^{8} - \frac{2}{9} a^{7} + \frac{4}{9} a^{6} + \frac{2}{9} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{117} a^{18} + \frac{1}{117} a^{17} - \frac{2}{117} a^{16} - \frac{28}{117} a^{13} + \frac{10}{117} a^{12} + \frac{11}{39} a^{11} + \frac{58}{117} a^{10} + \frac{7}{39} a^{9} - \frac{5}{13} a^{8} - \frac{49}{117} a^{7} + \frac{37}{117} a^{6} + \frac{17}{117} a^{5} + \frac{11}{117} a^{4} + \frac{2}{13} a^{3} + \frac{11}{39} a^{2} + \frac{4}{13} a + \frac{6}{13}$, $\frac{1}{4600055364167940004545027291} a^{19} - \frac{11084061679115971219493437}{4600055364167940004545027291} a^{18} + \frac{181027965684620241989838104}{4600055364167940004545027291} a^{17} + \frac{108310332843439522344324493}{4600055364167940004545027291} a^{16} + \frac{4585395106124877796044733}{117950137542767692424231469} a^{15} + \frac{10334282639096977054775552}{4600055364167940004545027291} a^{14} + \frac{12479811537974674907052301}{117950137542767692424231469} a^{13} - \frac{1271099129216882516158345421}{4600055364167940004545027291} a^{12} + \frac{2088211824079098899373325096}{4600055364167940004545027291} a^{11} - \frac{14615749563134226918514130}{4600055364167940004545027291} a^{10} - \frac{43758356771141977400282840}{1533351788055980001515009097} a^{9} + \frac{1148463643893033335619990887}{4600055364167940004545027291} a^{8} + \frac{12426192232192905150186793}{39316712514255897474743823} a^{7} - \frac{43157598274386123347504401}{1533351788055980001515009097} a^{6} + \frac{297507267429918471919789957}{4600055364167940004545027291} a^{5} - \frac{1636074402152640245407904635}{4600055364167940004545027291} a^{4} - \frac{428718677798773615104318725}{1533351788055980001515009097} a^{3} - \frac{508862452191614186313795470}{1533351788055980001515009097} a^{2} + \frac{76645086114000791989389}{569807427742839093836867} a + \frac{169364762973166904547329930}{511117262685326667171669699}$

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:  \( 47319000.8862 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T347:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 104 conjugacy class representatives for t20n347 are not computed
Character table for t20n347 is not computed

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.10.10368641602001.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
401Data not computed