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Properties

Label 20.16.2450305023...1984.1
Degree $20$
Signature $[16, 2]$
Discriminant $2^{50}\cdot 31^{10}\cdot 227^{4}$
Root discriminant $93.21$
Ramified primes $2, 31, 227$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1025

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4178, -64768, 116368, 837312, -204528, -2870084, -1062072, 3361668, 2312296, -1333928, -1504642, -70732, 295185, 106828, 14440, -12, -765, -116, -8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 8*x^18 - 116*x^17 - 765*x^16 - 12*x^15 + 14440*x^14 + 106828*x^13 + 295185*x^12 - 70732*x^11 - 1504642*x^10 - 1333928*x^9 + 2312296*x^8 + 3361668*x^7 - 1062072*x^6 - 2870084*x^5 - 204528*x^4 + 837312*x^3 + 116368*x^2 - 64768*x + 4178)
 
gp: K = bnfinit(x^20 - 4*x^19 - 8*x^18 - 116*x^17 - 765*x^16 - 12*x^15 + 14440*x^14 + 106828*x^13 + 295185*x^12 - 70732*x^11 - 1504642*x^10 - 1333928*x^9 + 2312296*x^8 + 3361668*x^7 - 1062072*x^6 - 2870084*x^5 - 204528*x^4 + 837312*x^3 + 116368*x^2 - 64768*x + 4178, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 8 x^{18} - 116 x^{17} - 765 x^{16} - 12 x^{15} + 14440 x^{14} + 106828 x^{13} + 295185 x^{12} - 70732 x^{11} - 1504642 x^{10} - 1333928 x^{9} + 2312296 x^{8} + 3361668 x^{7} - 1062072 x^{6} - 2870084 x^{5} - 204528 x^{4} + 837312 x^{3} + 116368 x^{2} - 64768 x + 4178 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2450305023038293482939005059275616681984=2^{50}\cdot 31^{10}\cdot 227^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $93.21$
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:  $2, 31, 227$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{167635797601132549917883392469076346947827754773704298134193} a^{19} + \frac{53272895912521600003829095237586963452254702702871065677945}{167635797601132549917883392469076346947827754773704298134193} a^{18} + \frac{49508707652022947416188360082568513019450544718970757541547}{167635797601132549917883392469076346947827754773704298134193} a^{17} + \frac{54836654231810993563789741673789503068283523512429279106656}{167635797601132549917883392469076346947827754773704298134193} a^{16} - \frac{21231750848812187667237690117386860094922932178560363266400}{167635797601132549917883392469076346947827754773704298134193} a^{15} - \frac{20922029259904474407945574250211516472923413221394385326945}{167635797601132549917883392469076346947827754773704298134193} a^{14} + \frac{58800364400372885327396734495483320467189580092248678942068}{167635797601132549917883392469076346947827754773704298134193} a^{13} + \frac{7348949105405522271589142468829928877124912512551045871861}{167635797601132549917883392469076346947827754773704298134193} a^{12} + \frac{30463062386211092643186943785744925921533118895265190202259}{167635797601132549917883392469076346947827754773704298134193} a^{11} - \frac{9751358814479005921943865549913171925675372741114713463139}{167635797601132549917883392469076346947827754773704298134193} a^{10} - \frac{5056441323487406397964639809530895512739790802637788290357}{167635797601132549917883392469076346947827754773704298134193} a^{9} - \frac{41938615510523265840301117289260065154159795801875273012433}{167635797601132549917883392469076346947827754773704298134193} a^{8} - \frac{51836517301086821968238339779271682515567951834907662898455}{167635797601132549917883392469076346947827754773704298134193} a^{7} - \frac{56887316639764344192393148807029760185699048463550982559418}{167635797601132549917883392469076346947827754773704298134193} a^{6} + \frac{50306605930535045927473438460874546046954388089674398503306}{167635797601132549917883392469076346947827754773704298134193} a^{5} - \frac{65760234076935678455338450123764516092956021137597409643610}{167635797601132549917883392469076346947827754773704298134193} a^{4} - \frac{5159322117482856964911684395941615929504006643803686626880}{167635797601132549917883392469076346947827754773704298134193} a^{3} + \frac{55950574357590863360197478243208654245108275987407136517804}{167635797601132549917883392469076346947827754773704298134193} a^{2} - \frac{78722348091766969626122511553415132269683545523713864925472}{167635797601132549917883392469076346947827754773704298134193} a + \frac{6914297027551742073152075701521285495781109659129337160604}{167635797601132549917883392469076346947827754773704298134193}$

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:  $17$
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:  \( 16330595940100 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1025:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 216 conjugacy class representatives for t20n1025 are not computed
Character table for t20n1025 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.207699287474176.1

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

Sibling fields

Degree 20 sibling: 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 $16{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ $16{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ $16{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ $16{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.22.7$x^{8} + 2 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
2.12.28.57$x^{12} + 2 x^{10} - 2 x^{8} + 4 x^{5} - 2 x^{4} - 2$$12$$1$$28$12T48$[2, 8/3, 8/3, 3]_{3}^{2}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.6.5.5$x^{6} + 10633$$6$$1$$5$$C_6$$[\ ]_{6}$
31.10.5.1$x^{10} - 1922 x^{6} + 923521 x^{2} - 2862915100$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
227Data not computed