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Properties

Label 20.0.29133682277...7536.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 3^{10}\cdot 11^{16}$
Root discriminant $33.36$
Ramified primes $2, 3, 11$
Class number $25$ (GRH)
Class group $[5, 5]$ (GRH)
Galois group $D_{10}$ (as 20T4)

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magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1024, -5120, 15232, -30336, 44688, -47040, 33296, -8320, -10492, 14984, -6028, -1612, 4033, -1982, 469, 158, -70, 26, 7, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 7*x^18 + 26*x^17 - 70*x^16 + 158*x^15 + 469*x^14 - 1982*x^13 + 4033*x^12 - 1612*x^11 - 6028*x^10 + 14984*x^9 - 10492*x^8 - 8320*x^7 + 33296*x^6 - 47040*x^5 + 44688*x^4 - 30336*x^3 + 15232*x^2 - 5120*x + 1024)
 
gp: K = bnfinit(x^20 - 2*x^19 + 7*x^18 + 26*x^17 - 70*x^16 + 158*x^15 + 469*x^14 - 1982*x^13 + 4033*x^12 - 1612*x^11 - 6028*x^10 + 14984*x^9 - 10492*x^8 - 8320*x^7 + 33296*x^6 - 47040*x^5 + 44688*x^4 - 30336*x^3 + 15232*x^2 - 5120*x + 1024, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 7 x^{18} + 26 x^{17} - 70 x^{16} + 158 x^{15} + 469 x^{14} - 1982 x^{13} + 4033 x^{12} - 1612 x^{11} - 6028 x^{10} + 14984 x^{9} - 10492 x^{8} - 8320 x^{7} + 33296 x^{6} - 47040 x^{5} + 44688 x^{4} - 30336 x^{3} + 15232 x^{2} - 5120 x + 1024 \)

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:  \(2913368227796180298080018497536=2^{30}\cdot 3^{10}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.36$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{8} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{6} a^{2} - \frac{1}{3} a$, $\frac{1}{6} a^{9} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{6} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{18} a^{11} - \frac{1}{18} a^{10} - \frac{1}{18} a^{9} + \frac{1}{18} a^{8} + \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{6} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{5}{18} a^{2} + \frac{1}{9} a + \frac{1}{9}$, $\frac{1}{36} a^{12} + \frac{1}{36} a^{10} - \frac{1}{18} a^{7} + \frac{5}{36} a^{6} + \frac{1}{12} a^{4} + \frac{5}{18} a^{3} + \frac{1}{6} a^{2} + \frac{1}{9} a + \frac{2}{9}$, $\frac{1}{72} a^{13} + \frac{1}{72} a^{11} - \frac{1}{36} a^{8} + \frac{5}{72} a^{7} + \frac{1}{24} a^{5} - \frac{13}{36} a^{4} - \frac{5}{12} a^{3} - \frac{4}{9} a^{2} - \frac{7}{18} a$, $\frac{1}{144} a^{14} + \frac{1}{144} a^{12} + \frac{5}{72} a^{9} + \frac{5}{144} a^{8} - \frac{1}{6} a^{7} - \frac{7}{48} a^{6} - \frac{13}{72} a^{5} + \frac{7}{24} a^{4} - \frac{5}{36} a^{3} - \frac{1}{36} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{144} a^{15} - \frac{1}{144} a^{13} - \frac{1}{72} a^{11} + \frac{5}{72} a^{10} + \frac{5}{144} a^{9} + \frac{1}{36} a^{8} + \frac{17}{144} a^{7} + \frac{11}{72} a^{6} - \frac{1}{12} a^{5} + \frac{2}{9} a^{4} - \frac{5}{18} a^{3} - \frac{2}{9} a^{2} + \frac{1}{18} a - \frac{1}{3}$, $\frac{1}{288} a^{16} - \frac{1}{288} a^{14} - \frac{1}{144} a^{12} - \frac{1}{48} a^{11} + \frac{7}{96} a^{10} + \frac{5}{72} a^{9} + \frac{1}{288} a^{8} - \frac{5}{144} a^{7} - \frac{11}{72} a^{6} + \frac{5}{18} a^{5} + \frac{13}{36} a^{4} + \frac{7}{18} a^{3} - \frac{7}{36} a^{2} - \frac{5}{18} a - \frac{1}{9}$, $\frac{1}{864} a^{17} + \frac{1}{864} a^{16} + \frac{1}{864} a^{15} - \frac{1}{288} a^{14} + \frac{1}{144} a^{12} - \frac{17}{864} a^{11} + \frac{61}{864} a^{10} - \frac{5}{864} a^{9} - \frac{17}{288} a^{8} - \frac{1}{18} a^{7} + \frac{7}{144} a^{6} - \frac{11}{27} a^{5} + \frac{23}{216} a^{4} - \frac{35}{108} a^{3} - \frac{2}{27} a^{2} - \frac{5}{27} a + \frac{4}{27}$, $\frac{1}{238464} a^{18} - \frac{17}{39744} a^{17} - \frac{107}{79488} a^{16} - \frac{113}{119232} a^{15} + \frac{11}{4416} a^{14} - \frac{271}{39744} a^{13} + \frac{205}{238464} a^{12} - \frac{635}{39744} a^{11} + \frac{961}{26496} a^{10} - \frac{541}{14904} a^{9} + \frac{145}{6624} a^{8} + \frac{155}{9936} a^{7} + \frac{389}{59616} a^{6} - \frac{49}{2484} a^{5} - \frac{1}{1656} a^{4} - \frac{265}{828} a^{3} - \frac{1961}{4968} a^{2} + \frac{65}{207} a + \frac{635}{1863}$, $\frac{1}{1070317215197584128} a^{19} - \frac{38884761835}{31479918094046592} a^{18} - \frac{197570772294091}{356772405065861376} a^{17} - \frac{502161184604405}{535158607598792064} a^{16} + \frac{830220264890753}{535158607598792064} a^{15} + \frac{543006647296517}{178386202532930688} a^{14} + \frac{5354894985993373}{1070317215197584128} a^{13} + \frac{3162400085228863}{535158607598792064} a^{12} + \frac{2335331214675187}{356772405065861376} a^{11} - \frac{6270865677276671}{133789651899698016} a^{10} - \frac{6955723438366091}{267579303799396032} a^{9} - \frac{661340468255219}{22298275316616336} a^{8} + \frac{30671786310835445}{267579303799396032} a^{7} + \frac{2986987108629581}{33447412974924504} a^{6} + \frac{1889964669559913}{22298275316616336} a^{5} - \frac{648291508419805}{3716379219436056} a^{4} + \frac{9656058395044375}{22298275316616336} a^{3} + \frac{690644951604113}{5574568829154084} a^{2} + \frac{1732475149384253}{8361853243731126} a - \frac{1232610926398864}{4180926621865563}$

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

Class group and class number

$C_{5}\times C_{5}$, which has order $25$ (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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{409480952879537}{535158607598792064} a^{19} + \frac{37875422087773}{31479918094046592} a^{18} - \frac{865114627388887}{178386202532930688} a^{17} - \frac{11708058333851869}{535158607598792064} a^{16} + \frac{5899847381939791}{133789651899698016} a^{15} - \frac{1136328791475917}{11149137658308168} a^{14} - \frac{9297553666603501}{23267765547773568} a^{13} + \frac{717975476197849013}{535158607598792064} a^{12} - \frac{19487670070485943}{7755921849257856} a^{11} + \frac{114223997820806759}{535158607598792064} a^{10} + \frac{6642034401318371}{1454235346735848} a^{9} - \frac{419830805751312569}{44596550633232672} a^{8} + \frac{551911637362985477}{133789651899698016} a^{7} + \frac{1017349902802518331}{133789651899698016} a^{6} - \frac{244084805712978685}{11149137658308168} a^{5} + \frac{33118638129474887}{1238793073145352} a^{4} - \frac{265788540198774887}{11149137658308168} a^{3} + \frac{160102196369469401}{11149137658308168} a^{2} - \frac{30892649327449195}{4180926621865563} a + \frac{9930119948685952}{4180926621865563} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4531870.2466 \) (assuming GRH)
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

\(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{2}, \sqrt{-3})\), 5.1.8433216.2 x5, 10.0.1706859170463744.3, 10.2.568953056821248.1 x5, 10.0.213357396307968.2 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 R R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$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}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$