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Properties

Label 20.4.30694870289...3125.2
Degree $20$
Signature $[4, 8]$
Discriminant $3^{10}\cdot 5^{11}\cdot 239^{8}$
Root discriminant $37.53$
Ramified primes $3, 5, 239$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T144

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-81, -1134, -2916, -5130, 22104, 7092, -47694, 17838, 10976, -13113, 4295, 2381, -4229, 2565, -907, 122, 71, -54, 23, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 23*x^18 - 54*x^17 + 71*x^16 + 122*x^15 - 907*x^14 + 2565*x^13 - 4229*x^12 + 2381*x^11 + 4295*x^10 - 13113*x^9 + 10976*x^8 + 17838*x^7 - 47694*x^6 + 7092*x^5 + 22104*x^4 - 5130*x^3 - 2916*x^2 - 1134*x - 81)
 
gp: K = bnfinit(x^20 - 5*x^19 + 23*x^18 - 54*x^17 + 71*x^16 + 122*x^15 - 907*x^14 + 2565*x^13 - 4229*x^12 + 2381*x^11 + 4295*x^10 - 13113*x^9 + 10976*x^8 + 17838*x^7 - 47694*x^6 + 7092*x^5 + 22104*x^4 - 5130*x^3 - 2916*x^2 - 1134*x - 81, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 23 x^{18} - 54 x^{17} + 71 x^{16} + 122 x^{15} - 907 x^{14} + 2565 x^{13} - 4229 x^{12} + 2381 x^{11} + 4295 x^{10} - 13113 x^{9} + 10976 x^{8} + 17838 x^{7} - 47694 x^{6} + 7092 x^{5} + 22104 x^{4} - 5130 x^{3} - 2916 x^{2} - 1134 x - 81 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(30694870289571932375789501953125=3^{10}\cdot 5^{11}\cdot 239^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.53$
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:  $3, 5, 239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{15} - \frac{1}{9} a^{14} + \frac{2}{9} a^{12} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{3} a^{9} + \frac{4}{9} a^{8} + \frac{2}{9} a^{7} + \frac{2}{9} a^{6} - \frac{1}{3} a^{5} - \frac{1}{9} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{99} a^{17} - \frac{14}{99} a^{15} - \frac{2}{99} a^{14} - \frac{4}{99} a^{13} - \frac{1}{33} a^{12} - \frac{2}{33} a^{11} + \frac{46}{99} a^{10} + \frac{37}{99} a^{9} - \frac{1}{9} a^{8} + \frac{8}{33} a^{7} - \frac{20}{99} a^{6} - \frac{2}{9} a^{5} - \frac{32}{99} a^{4} + \frac{1}{11} a^{3} + \frac{3}{11} a^{2} - \frac{3}{11} a - \frac{2}{11}$, $\frac{1}{6831} a^{18} - \frac{1}{621} a^{17} + \frac{140}{6831} a^{16} - \frac{140}{2277} a^{15} + \frac{623}{6831} a^{14} - \frac{817}{6831} a^{13} + \frac{1820}{6831} a^{12} + \frac{18}{253} a^{11} + \frac{125}{297} a^{10} - \frac{77}{621} a^{9} + \frac{959}{6831} a^{8} - \frac{364}{759} a^{7} - \frac{221}{621} a^{6} + \frac{587}{2277} a^{5} - \frac{95}{253} a^{4} + \frac{80}{759} a^{3} - \frac{47}{759} a^{2} + \frac{91}{253} a + \frac{8}{23}$, $\frac{1}{2559548842146441164029039901983226361807} a^{19} + \frac{36054059924251894475289556012564226}{853182947382147054676346633994408787269} a^{18} + \frac{5305266216588281802071822714361063925}{2559548842146441164029039901983226361807} a^{17} + \frac{137338957664870533949920641241884699685}{2559548842146441164029039901983226361807} a^{16} - \frac{303349024515631721192150033311858288270}{2559548842146441164029039901983226361807} a^{15} + \frac{4426486646976801970130524258006751377}{37094910755745524116362897130191686403} a^{14} + \frac{7945532223804985303372250016863475943}{37094910755745524116362897130191686403} a^{13} - \frac{66916817705413264801792496153032833691}{232686258376949196729912718362111487437} a^{12} - \frac{1017514133574521794244781802598717183189}{2559548842146441164029039901983226361807} a^{11} + \frac{853165532986007800117946680506548433961}{2559548842146441164029039901983226361807} a^{10} + \frac{13363043146329227179058516136295521638}{284394315794049018225448877998136262423} a^{9} + \frac{57845864984662421013525987938355821693}{232686258376949196729912718362111487437} a^{8} - \frac{861052728469352081244788232588826065454}{2559548842146441164029039901983226361807} a^{7} - \frac{910873747071601787402921198210426192750}{2559548842146441164029039901983226361807} a^{6} - \frac{9152708880673287597930236509658161329}{94798105264683006075149625999378754141} a^{5} + \frac{106531179538811990139049665008332673999}{853182947382147054676346633994408787269} a^{4} + \frac{81766391566994719481895853995757347119}{284394315794049018225448877998136262423} a^{3} + \frac{35343145861942617437515808714858419379}{94798105264683006075149625999378754141} a^{2} + \frac{35048748260037169110599332475800169547}{94798105264683006075149625999378754141} a - \frac{44944699647990405071805801979698489287}{94798105264683006075149625999378754141}$

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:  $11$
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:  \( 29399473.2143 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T144:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 40 conjugacy class representatives for t20n144
Character table for t20n144 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.12852225.1, 10.10.825898437253125.1

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 $20$ R R $20$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ $20$ $20$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
239Data not computed