Properties

Label 20.0.34078222410...2929.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 467^{2}\cdot 514417^{2}$
Root discriminant $11.93$
Ramified primes $3, 467, 514417$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois Group 20T1021

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 6, -11, 16, -17, 21, -22, 16, -14, 13, -6, 5, -7, 4, -4, 6, -4, 2, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 2*x^18 - 4*x^17 + 6*x^16 - 4*x^15 + 4*x^14 - 7*x^13 + 5*x^12 - 6*x^11 + 13*x^10 - 14*x^9 + 16*x^8 - 22*x^7 + 21*x^6 - 17*x^5 + 16*x^4 - 11*x^3 + 6*x^2 - 3*x + 1)
gp: K = bnfinit(x^20 - 2*x^19 + 2*x^18 - 4*x^17 + 6*x^16 - 4*x^15 + 4*x^14 - 7*x^13 + 5*x^12 - 6*x^11 + 13*x^10 - 14*x^9 + 16*x^8 - 22*x^7 + 21*x^6 - 17*x^5 + 16*x^4 - 11*x^3 + 6*x^2 - 3*x + 1, 1)

Normalized defining polynomial

\(x^{20} \) \(\mathstrut -\mathstrut 2 x^{19} \) \(\mathstrut +\mathstrut 2 x^{18} \) \(\mathstrut -\mathstrut 4 x^{17} \) \(\mathstrut +\mathstrut 6 x^{16} \) \(\mathstrut -\mathstrut 4 x^{15} \) \(\mathstrut +\mathstrut 4 x^{14} \) \(\mathstrut -\mathstrut 7 x^{13} \) \(\mathstrut +\mathstrut 5 x^{12} \) \(\mathstrut -\mathstrut 6 x^{11} \) \(\mathstrut +\mathstrut 13 x^{10} \) \(\mathstrut -\mathstrut 14 x^{9} \) \(\mathstrut +\mathstrut 16 x^{8} \) \(\mathstrut -\mathstrut 22 x^{7} \) \(\mathstrut +\mathstrut 21 x^{6} \) \(\mathstrut -\mathstrut 17 x^{5} \) \(\mathstrut +\mathstrut 16 x^{4} \) \(\mathstrut -\mathstrut 11 x^{3} \) \(\mathstrut +\mathstrut 6 x^{2} \) \(\mathstrut -\mathstrut 3 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(3407822241034569802929=3^{10}\cdot 467^{2}\cdot 514417^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.93$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 467, 514417$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{6197} a^{19} + \frac{117}{6197} a^{18} + \frac{1531}{6197} a^{17} + \frac{2472}{6197} a^{16} + \frac{2915}{6197} a^{15} - \frac{151}{6197} a^{14} + \frac{626}{6197} a^{13} + \frac{123}{6197} a^{12} + \frac{2248}{6197} a^{11} + \frac{1035}{6197} a^{10} - \frac{762}{6197} a^{9} + \frac{2263}{6197} a^{8} + \frac{2842}{6197} a^{7} - \frac{2659}{6197} a^{6} - \frac{353}{6197} a^{5} + \frac{1355}{6197} a^{4} + \frac{139}{6197} a^{3} - \frac{2061}{6197} a^{2} + \frac{2627}{6197} a + \frac{2760}{6197}$

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

Class group and class number

Trivial Abelian group, 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:  $9$
magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
Torsion generator:  \( \frac{5521}{6197} a^{19} - \frac{10925}{6197} a^{18} + \frac{12337}{6197} a^{17} - \frac{22670}{6197} a^{16} + \frac{31091}{6197} a^{15} - \frac{21864}{6197} a^{14} + \frac{23008}{6197} a^{13} - \frac{33572}{6197} a^{12} + \frac{23405}{6197} a^{11} - \frac{36581}{6197} a^{10} + \frac{68928}{6197} a^{9} - \frac{73493}{6197} a^{8} + \frac{92833}{6197} a^{7} - \frac{117389}{6197} a^{6} + \frac{108491}{6197} a^{5} - \frac{91779}{6197} a^{4} + \frac{79552}{6197} a^{3} - \frac{44468}{6197} a^{2} + \frac{27475}{6197} a - \frac{6660}{6197} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{1127}{6197} a^{19} - \frac{4475}{6197} a^{18} + \frac{2671}{6197} a^{17} - \frac{2706}{6197} a^{16} + \frac{13189}{6197} a^{15} - \frac{9055}{6197} a^{14} - \frac{956}{6197} a^{13} - \frac{16304}{6197} a^{12} + \frac{17514}{6197} a^{11} + \frac{1409}{6197} a^{10} + \frac{21200}{6197} a^{9} - \frac{33748}{6197} a^{8} + \frac{11479}{6197} a^{7} - \frac{34527}{6197} a^{6} + \frac{48353}{6197} a^{5} - \frac{22165}{6197} a^{4} + \frac{26516}{6197} a^{3} - \frac{29857}{6197} a^{2} + \frac{10857}{6197} a - \frac{6571}{6197} \),  \( a \),  \( \frac{432}{6197} a^{19} + \frac{968}{6197} a^{18} - \frac{1687}{6197} a^{17} - \frac{4177}{6197} a^{16} + \frac{1289}{6197} a^{15} + \frac{2935}{6197} a^{14} + \frac{10158}{6197} a^{13} - \frac{8834}{6197} a^{12} - \frac{7990}{6197} a^{11} - \frac{5261}{6197} a^{10} + \frac{11651}{6197} a^{9} + \frac{10884}{6197} a^{8} + \frac{6935}{6197} a^{7} - \frac{20834}{6197} a^{6} + \frac{2429}{6197} a^{5} - \frac{15749}{6197} a^{4} + \frac{22866}{6197} a^{3} - \frac{10378}{6197} a^{2} + \frac{7010}{6197} a - \frac{9898}{6197} \),  \( \frac{3887}{6197} a^{19} - \frac{3799}{6197} a^{18} + \frac{1877}{6197} a^{17} - \frac{9080}{6197} a^{16} + \frac{8686}{6197} a^{15} + \frac{1778}{6197} a^{14} + \frac{4038}{6197} a^{13} - \frac{11462}{6197} a^{12} + \frac{206}{6197} a^{11} - \frac{11202}{6197} a^{10} + \frac{25060}{6197} a^{9} - \frac{15853}{6197} a^{8} + \frac{22391}{6197} a^{7} - \frac{23725}{6197} a^{6} + \frac{16017}{6197} a^{5} - \frac{6762}{6197} a^{4} + \frac{7351}{6197} a^{3} + \frac{1614}{6197} a^{2} - \frac{1507}{6197} a + \frac{7310}{6197} \),  \( \frac{1795}{6197} a^{19} + \frac{5514}{6197} a^{18} - \frac{9520}{6197} a^{17} + \frac{188}{6197} a^{16} - \frac{10237}{6197} a^{15} + \frac{20214}{6197} a^{14} + \frac{2013}{6197} a^{13} - \frac{2307}{6197} a^{12} - \frac{23875}{6197} a^{11} + \frac{4922}{6197} a^{10} - \frac{10647}{6197} a^{9} + \frac{46429}{6197} a^{8} - \frac{23529}{6197} a^{7} + \frac{29770}{6197} a^{6} - \frac{51117}{6197} a^{5} + \frac{27789}{6197} a^{4} - \frac{10769}{6197} a^{3} + \frac{18705}{6197} a^{2} - \frac{452}{6197} a + \frac{2797}{6197} \),  \( \frac{2816}{6197} a^{19} - \frac{11363}{6197} a^{18} + \frac{10578}{6197} a^{17} - \frac{10473}{6197} a^{16} + \frac{28600}{6197} a^{15} - \frac{22411}{6197} a^{14} + \frac{2868}{6197} a^{13} - \frac{25452}{6197} a^{12} + \frac{34216}{6197} a^{11} - \frac{10424}{6197} a^{10} + \frac{47946}{6197} a^{9} - \frac{78469}{6197} a^{8} + \frac{52321}{6197} a^{7} - \frac{82329}{6197} a^{6} + \frac{96624}{6197} a^{5} - \frac{57445}{6197} a^{4} + \frac{44392}{6197} a^{3} - \frac{40566}{6197} a^{2} + \frac{10808}{6197} a + \frac{1122}{6197} \),  \( \frac{5768}{6197} a^{19} - \frac{6814}{6197} a^{18} + \frac{6280}{6197} a^{17} - \frac{19392}{6197} a^{16} + \frac{19850}{6197} a^{15} - \frac{9585}{6197} a^{14} + \frac{22705}{6197} a^{13} - \frac{27979}{6197} a^{12} + \frac{8537}{6197} a^{11} - \frac{35013}{6197} a^{10} + \frac{54230}{6197} a^{9} - \frac{35080}{6197} a^{8} + \frac{69758}{6197} a^{7} - \frac{86295}{6197} a^{6} + \frac{64679}{6197} a^{5} - \frac{66944}{6197} a^{4} + \frac{64309}{6197} a^{3} - \frac{32987}{6197} a^{2} + \frac{25659}{6197} a - \frac{6610}{6197} \),  \( \frac{8412}{6197} a^{19} - \frac{13513}{6197} a^{18} + \frac{13800}{6197} a^{17} - \frac{33653}{6197} a^{16} + \frac{42830}{6197} a^{15} - \frac{24615}{6197} a^{14} + \frac{35644}{6197} a^{13} - \frac{49799}{6197} a^{12} + \frac{21720}{6197} a^{11} - \frac{49941}{6197} a^{10} + \frac{96906}{6197} a^{9} - \frac{87586}{6197} a^{8} + \frac{129015}{6197} a^{7} - \frac{169854}{6197} a^{6} + \frac{135261}{6197} a^{5} - \frac{121963}{6197} a^{4} + \frac{115778}{6197} a^{3} - \frac{53699}{6197} a^{2} + \frac{37004}{6197} a - \frac{15433}{6197} \),  \( \frac{2066}{6197} a^{19} + \frac{39}{6197} a^{18} - \frac{3621}{6197} a^{17} + \frac{824}{6197} a^{16} - \frac{7291}{6197} a^{15} + \frac{16475}{6197} a^{14} - \frac{8054}{6197} a^{13} + \frac{6238}{6197} a^{12} - \frac{15776}{6197} a^{11} + \frac{345}{6197} a^{10} - \frac{254}{6197} a^{9} + \frac{27608}{6197} a^{8} - \frac{21775}{6197} a^{7} + \frac{28033}{6197} a^{6} - \frac{53825}{6197} a^{5} + \frac{47962}{6197} a^{4} - \frac{35070}{6197} a^{3} + \frac{36495}{6197} a^{2} - \frac{19781}{6197} a + \frac{7117}{6197} \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 658.849285992 \) (assuming GRH)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

20T1021:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 7257600
The 84 conjugacy class representatives for t20n1021 are not computed
Character table for t20n1021 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 10.0.240232739.1

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 sibling: 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 ${\href{/LocalNumberField/5.14.0.1}{14} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $18{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
467Data not computed
514417Data not computed