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Properties

Label 20.6.78574805586...9483.1
Degree $20$
Signature $[6, 7]$
Discriminant $-\,13^{10}\cdot 97^{2}\cdot 347^{7}$
Root discriminant $44.13$
Ramified primes $13, 97, 347$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T796

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![729661, -3835220, 4345303, -2061817, 1545640, -32234, -1521385, 1093654, -158699, -178945, 122424, -15033, -20751, 12031, -3651, 173, 242, -113, 36, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 36*x^18 - 113*x^17 + 242*x^16 + 173*x^15 - 3651*x^14 + 12031*x^13 - 20751*x^12 - 15033*x^11 + 122424*x^10 - 178945*x^9 - 158699*x^8 + 1093654*x^7 - 1521385*x^6 - 32234*x^5 + 1545640*x^4 - 2061817*x^3 + 4345303*x^2 - 3835220*x + 729661)
 
gp: K = bnfinit(x^20 - 6*x^19 + 36*x^18 - 113*x^17 + 242*x^16 + 173*x^15 - 3651*x^14 + 12031*x^13 - 20751*x^12 - 15033*x^11 + 122424*x^10 - 178945*x^9 - 158699*x^8 + 1093654*x^7 - 1521385*x^6 - 32234*x^5 + 1545640*x^4 - 2061817*x^3 + 4345303*x^2 - 3835220*x + 729661, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 36 x^{18} - 113 x^{17} + 242 x^{16} + 173 x^{15} - 3651 x^{14} + 12031 x^{13} - 20751 x^{12} - 15033 x^{11} + 122424 x^{10} - 178945 x^{9} - 158699 x^{8} + 1093654 x^{7} - 1521385 x^{6} - 32234 x^{5} + 1545640 x^{4} - 2061817 x^{3} + 4345303 x^{2} - 3835220 x + 729661 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-785748055861331850537130300409483=-\,13^{10}\cdot 97^{2}\cdot 347^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.13$
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:  $13, 97, 347$
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}{6119333743402954946034168394174396317919863670542874843941916986796607} a^{19} - \frac{590483715839564737747360216773086251398730688474741479915196944827698}{6119333743402954946034168394174396317919863670542874843941916986796607} a^{18} + \frac{194411277420290921902084462288812347250007764667251911417287145398095}{6119333743402954946034168394174396317919863670542874843941916986796607} a^{17} - \frac{2483891482922521355847924344605103843381829133598684125653271759545172}{6119333743402954946034168394174396317919863670542874843941916986796607} a^{16} - \frac{1952235830152991470155924293203270835565292519514128918429141808970327}{6119333743402954946034168394174396317919863670542874843941916986796607} a^{15} + \frac{1325717493851707127997213376674724049731448253851733129239397344480474}{6119333743402954946034168394174396317919863670542874843941916986796607} a^{14} - \frac{22044254367236285842001765357122674232209666820772883657003388048526}{6119333743402954946034168394174396317919863670542874843941916986796607} a^{13} + \frac{3035551486305778652045017820576601502462328604721535261220397331198344}{6119333743402954946034168394174396317919863670542874843941916986796607} a^{12} + \frac{146627368922892939549614089470943532120204870762892612910920786810673}{556303067582086813275833490379490574356351242776624985812901544254237} a^{11} - \frac{2561698167731771035194637747413995395312949919717618325338587440435793}{6119333743402954946034168394174396317919863670542874843941916986796607} a^{10} + \frac{1675427823402896944214571522921650448249317169051254300768069018251810}{6119333743402954946034168394174396317919863670542874843941916986796607} a^{9} + \frac{2878240383204384925232596068961707514097459688198369957768920423280865}{6119333743402954946034168394174396317919863670542874843941916986796607} a^{8} - \frac{786035637682712717924105453634800245709903283577627890741281572253946}{6119333743402954946034168394174396317919863670542874843941916986796607} a^{7} - \frac{2781636756028320904888506361045755932410107051550274845757258642604508}{6119333743402954946034168394174396317919863670542874843941916986796607} a^{6} + \frac{2960246792322268875171195390529402186831732608508468566993927592368537}{6119333743402954946034168394174396317919863670542874843941916986796607} a^{5} - \frac{1266117982865865012918672303763511267653248373437770560123827332225699}{6119333743402954946034168394174396317919863670542874843941916986796607} a^{4} + \frac{264733971542990471711807966224280213835325320527612257290879638964802}{6119333743402954946034168394174396317919863670542874843941916986796607} a^{3} + \frac{2512574870403338856197598972987896048059515509282086286403594595709597}{6119333743402954946034168394174396317919863670542874843941916986796607} a^{2} + \frac{2579391812442795254415473862515982081385737536639212240289035550662402}{6119333743402954946034168394174396317919863670542874843941916986796607} a - \frac{2542716338275953574543096823174909689188739332259768446148918476346498}{6119333743402954946034168394174396317919863670542874843941916986796607}$

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:  $12$
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:  \( 390933084.808 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T796:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 108 conjugacy class representatives for t20n796 are not computed
Character table for t20n796 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.3.4511.1, 10.6.44707018837.1

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 $20$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
347Data not computed