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Properties

Label 20.0.15398252078...1801.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 7993^{6}$
Root discriminant $25.67$
Ramified primes $3, 7993$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T199

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -30, 189, -354, 717, -808, 1139, -964, 1155, -753, 848, -313, 480, -94, 184, -17, 52, -2, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 9*x^18 - 2*x^17 + 52*x^16 - 17*x^15 + 184*x^14 - 94*x^13 + 480*x^12 - 313*x^11 + 848*x^10 - 753*x^9 + 1155*x^8 - 964*x^7 + 1139*x^6 - 808*x^5 + 717*x^4 - 354*x^3 + 189*x^2 - 30*x + 4)
 
gp: K = bnfinit(x^20 + 9*x^18 - 2*x^17 + 52*x^16 - 17*x^15 + 184*x^14 - 94*x^13 + 480*x^12 - 313*x^11 + 848*x^10 - 753*x^9 + 1155*x^8 - 964*x^7 + 1139*x^6 - 808*x^5 + 717*x^4 - 354*x^3 + 189*x^2 - 30*x + 4, 1)
 

Normalized defining polynomial

\( x^{20} + 9 x^{18} - 2 x^{17} + 52 x^{16} - 17 x^{15} + 184 x^{14} - 94 x^{13} + 480 x^{12} - 313 x^{11} + 848 x^{10} - 753 x^{9} + 1155 x^{8} - 964 x^{7} + 1139 x^{6} - 808 x^{5} + 717 x^{4} - 354 x^{3} + 189 x^{2} - 30 x + 4 \)

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:  \(15398252078750514273032991801=3^{10}\cdot 7993^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.67$
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:  $3, 7993$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{13} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{16} - \frac{1}{9} a^{15} - \frac{1}{9} a^{14} + \frac{1}{3} a^{13} + \frac{2}{9} a^{12} - \frac{4}{9} a^{11} + \frac{1}{9} a^{9} - \frac{4}{9} a^{8} - \frac{4}{9} a^{7} + \frac{2}{9} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{4}{9} a^{3} - \frac{1}{3} a^{2} - \frac{2}{9} a - \frac{1}{9}$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{16} + \frac{1}{9} a^{14} - \frac{4}{9} a^{13} - \frac{2}{9} a^{11} + \frac{1}{9} a^{10} - \frac{2}{9} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{9} a^{6} + \frac{4}{9} a^{4} + \frac{2}{9} a^{3} + \frac{1}{9} a^{2} - \frac{2}{9} a + \frac{4}{9}$, $\frac{1}{3597097359404039886} a^{19} - \frac{40803402784458376}{1798548679702019943} a^{18} - \frac{152385093991330979}{3597097359404039886} a^{17} - \frac{84114307508210872}{1798548679702019943} a^{16} + \frac{234101528676993689}{1798548679702019943} a^{15} - \frac{39437450845603537}{399677484378226654} a^{14} - \frac{157175987412789656}{1798548679702019943} a^{13} + \frac{528889739498715554}{1798548679702019943} a^{12} - \frac{194975253983554304}{1798548679702019943} a^{11} - \frac{1025688603506725771}{3597097359404039886} a^{10} + \frac{342562365227380595}{1798548679702019943} a^{9} + \frac{581427653391720071}{1199032453134679962} a^{8} - \frac{971548885566900905}{3597097359404039886} a^{7} + \frac{390536742618783503}{1798548679702019943} a^{6} - \frac{1479263681319091955}{3597097359404039886} a^{5} + \frac{234421436864653841}{599516226567339981} a^{4} - \frac{450366240762007091}{1199032453134679962} a^{3} - \frac{615785159353798985}{1798548679702019943} a^{2} - \frac{1199262339560373703}{3597097359404039886} a + \frac{783703626342298760}{1798548679702019943}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( -\frac{129816885194035837}{3597097359404039886} a^{19} - \frac{8941188406976300}{1798548679702019943} a^{18} - \frac{395053856587338295}{1199032453134679962} a^{17} + \frac{17529178478014373}{599516226567339981} a^{16} - \frac{3432148424168746330}{1798548679702019943} a^{15} + \frac{1355896393469959955}{3597097359404039886} a^{14} - \frac{12225936668494984132}{1798548679702019943} a^{13} + \frac{4713409032326223905}{1798548679702019943} a^{12} - \frac{10618059553893811855}{599516226567339981} a^{11} + \frac{34204545468602823469}{3597097359404039886} a^{10} - \frac{18767015864170069910}{599516226567339981} a^{9} + \frac{88897099381875047165}{3597097359404039886} a^{8} - \frac{150817970512061193395}{3597097359404039886} a^{7} + \frac{58658453132482240670}{1798548679702019943} a^{6} - \frac{150247661532619819945}{3597097359404039886} a^{5} + \frac{16589949156055995478}{599516226567339981} a^{4} - \frac{95413669206221897165}{3597097359404039886} a^{3} + \frac{20883959034236605820}{1798548679702019943} a^{2} - \frac{27449170429242085297}{3597097359404039886} a + \frac{2193851988355898843}{1798548679702019943} \) (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:  \( 2509777.3261 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T199:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1440
The 22 conjugacy class representatives for t20n199
Character table for t20n199 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 10.4.13787743742739.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 20 siblings: data not computed
Degree 24 siblings: data not computed
Degree 30 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7993Data not computed