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Properties

Label 20.0.79356918283...1401.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 103^{10}$
Root discriminant $17.58$
Ramified primes $3, 103$
Class number $1$
Class group Trivial
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, -351, 828, -1833, 3313, -4105, 4120, -3379, 2561, -1756, 1059, -619, 359, -230, 108, -35, 26, -8, 1, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + x^18 - 8*x^17 + 26*x^16 - 35*x^15 + 108*x^14 - 230*x^13 + 359*x^12 - 619*x^11 + 1059*x^10 - 1756*x^9 + 2561*x^8 - 3379*x^7 + 4120*x^6 - 4105*x^5 + 3313*x^4 - 1833*x^3 + 828*x^2 - 351*x + 81)
 
gp: K = bnfinit(x^20 - 2*x^19 + x^18 - 8*x^17 + 26*x^16 - 35*x^15 + 108*x^14 - 230*x^13 + 359*x^12 - 619*x^11 + 1059*x^10 - 1756*x^9 + 2561*x^8 - 3379*x^7 + 4120*x^6 - 4105*x^5 + 3313*x^4 - 1833*x^3 + 828*x^2 - 351*x + 81, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + x^{18} - 8 x^{17} + 26 x^{16} - 35 x^{15} + 108 x^{14} - 230 x^{13} + 359 x^{12} - 619 x^{11} + 1059 x^{10} - 1756 x^{9} + 2561 x^{8} - 3379 x^{7} + 4120 x^{6} - 4105 x^{5} + 3313 x^{4} - 1833 x^{3} + 828 x^{2} - 351 x + 81 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7935691828389105528301401=3^{10}\cdot 103^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.58$
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:  $3, 103$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{15} a^{13} + \frac{1}{5} a^{12} - \frac{4}{15} a^{11} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{7}{15} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{4}{15} a^{3} + \frac{1}{5} a^{2} + \frac{7}{15} a - \frac{1}{5}$, $\frac{1}{15} a^{14} + \frac{2}{15} a^{12} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{4}{15} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{4}{15} a^{4} - \frac{2}{15} a^{2} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{15} a^{15} + \frac{2}{5} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{2}{5} a^{3} - \frac{1}{3} a + \frac{2}{5}$, $\frac{1}{45} a^{16} - \frac{1}{45} a^{15} + \frac{1}{45} a^{14} + \frac{1}{45} a^{13} + \frac{16}{45} a^{12} + \frac{4}{15} a^{11} + \frac{22}{45} a^{10} - \frac{2}{9} a^{9} - \frac{16}{45} a^{8} + \frac{4}{15} a^{7} - \frac{1}{45} a^{6} - \frac{14}{45} a^{5} + \frac{17}{45} a^{4} - \frac{1}{45} a^{3} + \frac{8}{45} a^{2} + \frac{4}{15} a + \frac{2}{5}$, $\frac{1}{45} a^{17} - \frac{1}{45} a^{14} - \frac{1}{45} a^{13} + \frac{13}{45} a^{12} - \frac{4}{9} a^{11} + \frac{7}{15} a^{10} - \frac{17}{45} a^{9} + \frac{11}{45} a^{8} - \frac{16}{45} a^{7} - \frac{2}{15} a^{6} + \frac{7}{15} a^{5} - \frac{17}{45} a^{4} - \frac{11}{45} a^{3} + \frac{17}{45} a^{2} + \frac{7}{15} a$, $\frac{1}{211725} a^{18} - \frac{108}{23525} a^{17} + \frac{59}{211725} a^{16} - \frac{38}{70575} a^{15} + \frac{763}{211725} a^{14} - \frac{209}{70575} a^{13} + \frac{13024}{70575} a^{12} - \frac{34192}{70575} a^{11} + \frac{5381}{14115} a^{10} - \frac{16279}{70575} a^{9} + \frac{9653}{70575} a^{8} + \frac{31606}{70575} a^{7} - \frac{59222}{211725} a^{6} - \frac{5092}{14115} a^{5} - \frac{95266}{211725} a^{4} + \frac{746}{4705} a^{3} + \frac{18263}{42345} a^{2} + \frac{10598}{23525} a + \frac{1402}{23525}$, $\frac{1}{46346922183209310225} a^{19} + \frac{97813867331131}{46346922183209310225} a^{18} - \frac{171324230973836072}{46346922183209310225} a^{17} + \frac{292416488968100698}{46346922183209310225} a^{16} - \frac{1426720111377446584}{46346922183209310225} a^{15} + \frac{1060954167381284707}{46346922183209310225} a^{14} - \frac{402202232019242473}{15448974061069770075} a^{13} + \frac{1627099606191312194}{9269384436641862045} a^{12} + \frac{7466979644993272637}{46346922183209310225} a^{11} - \frac{13334493211706875597}{46346922183209310225} a^{10} - \frac{483774090584182146}{1716552673452196675} a^{9} - \frac{2692613804506074737}{9269384436641862045} a^{8} + \frac{22240529300694180737}{46346922183209310225} a^{7} + \frac{7017832831469383409}{46346922183209310225} a^{6} + \frac{7697007007927846954}{46346922183209310225} a^{5} - \frac{20318561107488464308}{46346922183209310225} a^{4} + \frac{287540069107498621}{1853876887328372409} a^{3} - \frac{164515035435150134}{1716552673452196675} a^{2} + \frac{2564571920003835541}{5149658020356590025} a + \frac{821982136613316587}{1716552673452196675}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{740936717620306}{49252839727108725} a^{19} + \frac{1131241806051221}{49252839727108725} a^{18} - \frac{256037258732932}{49252839727108725} a^{17} + \frac{1173138861135319}{9850567945421745} a^{16} - \frac{16495279595571029}{49252839727108725} a^{15} + \frac{18526342157966534}{49252839727108725} a^{14} - \frac{4817673065088434}{3283522648473915} a^{13} + \frac{137291307444535049}{49252839727108725} a^{12} - \frac{205751485369155344}{49252839727108725} a^{11} + \frac{369054225223860652}{49252839727108725} a^{10} - \frac{207564361186390564}{16417613242369575} a^{9} + \frac{1029300758390361373}{49252839727108725} a^{8} - \frac{1449127045179420116}{49252839727108725} a^{7} + \frac{1881344828746211557}{49252839727108725} a^{6} - \frac{2250482827550976964}{49252839727108725} a^{5} + \frac{2090359118592799051}{49252839727108725} a^{4} - \frac{64051751454156340}{1970113589084349} a^{3} + \frac{240893505545010541}{16417613242369575} a^{2} - \frac{8629029729319543}{1094507549491305} a + \frac{5848002379526821}{1824179249152175} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 60789.2486151 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-103}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{309}) \), \(\Q(\sqrt{-3}, \sqrt{-103})\), 5.1.10609.1 x5, 10.0.11592740743.1, 10.0.27349864083.1 x5, 10.2.2817036000549.1 x5

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$103$103.2.1.2$x^{2} + 206$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.2$x^{2} + 206$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.2$x^{2} + 206$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.2$x^{2} + 206$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.2$x^{2} + 206$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.2$x^{2} + 206$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.2$x^{2} + 206$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.2$x^{2} + 206$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.2$x^{2} + 206$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.2$x^{2} + 206$$2$$1$$1$$C_2$$[\ ]_{2}$