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Properties

Label 20.0.18759175710...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{28}\cdot 5^{12}\cdot 17^{15}$
Root discriminant $58.03$
Ramified primes $2, 5, 17$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_4\times F_5$ (as 20T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![82460068, -10744784, 34888, 3558644, -2032044, 648684, -251414, 432020, -167032, 724, 15319, 7610, 5873, -4820, 536, -48, 238, -44, -1, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - x^18 - 44*x^17 + 238*x^16 - 48*x^15 + 536*x^14 - 4820*x^13 + 5873*x^12 + 7610*x^11 + 15319*x^10 + 724*x^9 - 167032*x^8 + 432020*x^7 - 251414*x^6 + 648684*x^5 - 2032044*x^4 + 3558644*x^3 + 34888*x^2 - 10744784*x + 82460068)
 
gp: K = bnfinit(x^20 - 2*x^19 - x^18 - 44*x^17 + 238*x^16 - 48*x^15 + 536*x^14 - 4820*x^13 + 5873*x^12 + 7610*x^11 + 15319*x^10 + 724*x^9 - 167032*x^8 + 432020*x^7 - 251414*x^6 + 648684*x^5 - 2032044*x^4 + 3558644*x^3 + 34888*x^2 - 10744784*x + 82460068, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - x^{18} - 44 x^{17} + 238 x^{16} - 48 x^{15} + 536 x^{14} - 4820 x^{13} + 5873 x^{12} + 7610 x^{11} + 15319 x^{10} + 724 x^{9} - 167032 x^{8} + 432020 x^{7} - 251414 x^{6} + 648684 x^{5} - 2032044 x^{4} + 3558644 x^{3} + 34888 x^{2} - 10744784 x + 82460068 \)

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:  \(187591757103747287810048000000000000=2^{28}\cdot 5^{12}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.03$
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:  $2, 5, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10}$, $\frac{1}{24} a^{15} + \frac{1}{24} a^{14} - \frac{1}{12} a^{13} + \frac{1}{12} a^{12} + \frac{5}{24} a^{11} + \frac{5}{24} a^{10} - \frac{1}{4} a^{9} + \frac{1}{12} a^{8} - \frac{1}{12} a^{7} - \frac{1}{12} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{2} a^{2} + \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{144} a^{16} + \frac{1}{72} a^{15} + \frac{5}{144} a^{14} + \frac{31}{144} a^{12} - \frac{1}{72} a^{11} - \frac{7}{144} a^{10} - \frac{1}{9} a^{9} + \frac{1}{12} a^{8} + \frac{1}{18} a^{7} + \frac{1}{8} a^{6} + \frac{13}{36} a^{5} + \frac{2}{9} a^{4} + \frac{1}{4} a^{3} + \frac{1}{9} a^{2} - \frac{4}{9} a + \frac{13}{36}$, $\frac{1}{144} a^{17} + \frac{1}{144} a^{15} - \frac{5}{72} a^{14} - \frac{5}{144} a^{13} - \frac{7}{36} a^{12} + \frac{11}{48} a^{11} + \frac{17}{72} a^{10} - \frac{7}{36} a^{9} - \frac{1}{9} a^{8} + \frac{1}{72} a^{7} - \frac{7}{18} a^{6} - \frac{7}{36} a^{4} - \frac{7}{18} a^{3} + \frac{1}{3} a^{2} + \frac{1}{4} a + \frac{5}{18}$, $\frac{1}{5472} a^{18} + \frac{1}{1368} a^{17} - \frac{1}{304} a^{16} + \frac{2}{171} a^{15} + \frac{55}{1368} a^{14} - \frac{43}{456} a^{13} + \frac{101}{684} a^{12} + \frac{11}{456} a^{11} - \frac{1307}{5472} a^{10} + \frac{121}{684} a^{9} - \frac{73}{2736} a^{8} + \frac{37}{342} a^{7} + \frac{293}{2736} a^{6} + \frac{89}{684} a^{5} - \frac{7}{36} a^{4} + \frac{289}{1368} a^{3} + \frac{377}{1368} a^{2} - \frac{41}{684} a - \frac{43}{152}$, $\frac{1}{10599432967902866089529631329713377077737015100258009627666112} a^{19} + \frac{297865976246194277952821835466208329637634703098797139257}{3533144322634288696509877109904459025912338366752669875888704} a^{18} + \frac{17384830145611190721435159577229392008082734817618209365317}{5299716483951433044764815664856688538868507550129004813833056} a^{17} - \frac{12558970168018964840188098647385103751422332586602458060801}{5299716483951433044764815664856688538868507550129004813833056} a^{16} + \frac{11956680112912531230065419002079937500539055006385139394883}{2649858241975716522382407832428344269434253775064502406916528} a^{15} + \frac{257975664375235297945415527126054788875655125956886854260589}{2649858241975716522382407832428344269434253775064502406916528} a^{14} + \frac{286344210489094905217441580022086681997075670508976235028563}{2649858241975716522382407832428344269434253775064502406916528} a^{13} - \frac{8461516359420556371777315853798207802927263466342324858273}{55205380041160760882966829842257172279880286980510466810761} a^{12} + \frac{1058013601511580504187497393142597023525043253922417713597729}{10599432967902866089529631329713377077737015100258009627666112} a^{11} + \frac{526973683216243982994914954250471100004061035624146956587189}{3533144322634288696509877109904459025912338366752669875888704} a^{10} + \frac{927214685023063915247876381668435664447539987229161182432491}{5299716483951433044764815664856688538868507550129004813833056} a^{9} + \frac{22433290092841334292355822033377021580574469921331708114529}{92977482174586544644996766050117342787166799125070259891808} a^{8} + \frac{1136418300745698335389247198676410233801338026667380736799981}{5299716483951433044764815664856688538868507550129004813833056} a^{7} + \frac{541289514911133403343095049835908709896254274945066785443673}{1766572161317144348254938554952229512956169183376334937944352} a^{6} + \frac{7431632886518477430382323536003082028222291325977815584465}{25479406172843428099830844542580233359944747837158676989582} a^{5} + \frac{1295189122626352934474459198852811778008665382250381468930235}{2649858241975716522382407832428344269434253775064502406916528} a^{4} - \frac{3902980997181452627973360149306028044752469427258856829075}{50958812345686856199661689085160466719889495674317353979164} a^{3} - \frac{508927331286275244557097908276592960557305761345911927973635}{2649858241975716522382407832428344269434253775064502406916528} a^{2} + \frac{47423825730413773677166413550479282727627941970558200437215}{294428693552857391375823092492038252159361530562722489657392} a + \frac{255991897280151091285635572924258990707090533860204302088407}{2649858241975716522382407832428344269434253775064502406916528}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( -1 \) (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:  \( 2581061275.0473027 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times F_5$ (as 20T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_4\times F_5$
Character table for $C_4\times F_5$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.0.78608.1, 5.1.578000.2, 10.2.5679428000000.1

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

Sibling fields

Degree 20 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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$2.10.14.5$x^{10} - 2 x^{5} - 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
2.10.14.5$x^{10} - 2 x^{5} - 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
5Data not computed
17Data not computed