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Properties

Label 20.4.19833781175...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{4}\cdot 5^{13}\cdot 6329^{5}$
Root discriminant $29.17$
Ramified primes $2, 5, 6329$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1037

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8804, -33464, 59290, -63712, 46207, -22521, 5233, 3098, -5690, 5226, -3040, 981, -164, 64, -61, 74, -72, 30, 2, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 2*x^18 + 30*x^17 - 72*x^16 + 74*x^15 - 61*x^14 + 64*x^13 - 164*x^12 + 981*x^11 - 3040*x^10 + 5226*x^9 - 5690*x^8 + 3098*x^7 + 5233*x^6 - 22521*x^5 + 46207*x^4 - 63712*x^3 + 59290*x^2 - 33464*x + 8804)
 
gp: K = bnfinit(x^20 - 5*x^19 + 2*x^18 + 30*x^17 - 72*x^16 + 74*x^15 - 61*x^14 + 64*x^13 - 164*x^12 + 981*x^11 - 3040*x^10 + 5226*x^9 - 5690*x^8 + 3098*x^7 + 5233*x^6 - 22521*x^5 + 46207*x^4 - 63712*x^3 + 59290*x^2 - 33464*x + 8804, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 2 x^{18} + 30 x^{17} - 72 x^{16} + 74 x^{15} - 61 x^{14} + 64 x^{13} - 164 x^{12} + 981 x^{11} - 3040 x^{10} + 5226 x^{9} - 5690 x^{8} + 3098 x^{7} + 5233 x^{6} - 22521 x^{5} + 46207 x^{4} - 63712 x^{3} + 59290 x^{2} - 33464 x + 8804 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(198337811759061770488281250000=2^{4}\cdot 5^{13}\cdot 6329^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.17$
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:  $2, 5, 6329$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{8562885583299084427561499753766576646} a^{19} + \frac{801618640930459260799557907290613734}{4281442791649542213780749876883288323} a^{18} + \frac{2380912990657514874510123490921992975}{8562885583299084427561499753766576646} a^{17} + \frac{2114730573420710540666670441799796543}{4281442791649542213780749876883288323} a^{16} + \frac{1044012983451462725749725134899801618}{4281442791649542213780749876883288323} a^{15} - \frac{64242811626494433080095556297396675}{4281442791649542213780749876883288323} a^{14} - \frac{2406937656139748979035305859360546385}{8562885583299084427561499753766576646} a^{13} - \frac{3293583729709419361209202269267380879}{8562885583299084427561499753766576646} a^{12} + \frac{1001668811409544042980760432024513533}{4281442791649542213780749876883288323} a^{11} + \frac{3387867221590503871901963924052641629}{8562885583299084427561499753766576646} a^{10} + \frac{3334375878238102858819035376460765897}{8562885583299084427561499753766576646} a^{9} + \frac{146185008252029952689766370267410156}{4281442791649542213780749876883288323} a^{8} - \frac{163101364275122134951128830928845828}{4281442791649542213780749876883288323} a^{7} + \frac{1243594223066683787345128194368651479}{4281442791649542213780749876883288323} a^{6} - \frac{2510166129358959667437211793597607377}{8562885583299084427561499753766576646} a^{5} + \frac{344343742191545812034119258796377764}{4281442791649542213780749876883288323} a^{4} + \frac{1497155981787079745925503423163571491}{4281442791649542213780749876883288323} a^{3} - \frac{868475220601712004166457460604954189}{8562885583299084427561499753766576646} a^{2} - \frac{1061107254079721768610477426752511736}{4281442791649542213780749876883288323} a + \frac{1716724183780442197763039438154576754}{4281442791649542213780749876883288323}$

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:  $11$
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:  \( 8410758.56288 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1037:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 384 conjugacy class representatives for t20n1037 are not computed
Character table for t20n1037 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.625878765625.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.8.0.1}{8} }$ R $16{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ $16{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.8.0.1}{8} }$ $16{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
6329Data not computed