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Properties

Label 20.14.5784566108...4375.1
Degree $20$
Signature $[14, 3]$
Discriminant $-\,5^{15}\cdot 11^{7}\cdot 9931^{4}$
Root discriminant $48.77$
Ramified primes $5, 11, 9931$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1023

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, 245, 670, -13455, -6530, 101860, -74560, -94170, 93845, 13490, -40015, 10395, 6196, -2555, 297, -28, -36, 42, -16, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 16*x^18 + 42*x^17 - 36*x^16 - 28*x^15 + 297*x^14 - 2555*x^13 + 6196*x^12 + 10395*x^11 - 40015*x^10 + 13490*x^9 + 93845*x^8 - 94170*x^7 - 74560*x^6 + 101860*x^5 - 6530*x^4 - 13455*x^3 + 670*x^2 + 245*x - 5)
 
gp: K = bnfinit(x^20 - 2*x^19 - 16*x^18 + 42*x^17 - 36*x^16 - 28*x^15 + 297*x^14 - 2555*x^13 + 6196*x^12 + 10395*x^11 - 40015*x^10 + 13490*x^9 + 93845*x^8 - 94170*x^7 - 74560*x^6 + 101860*x^5 - 6530*x^4 - 13455*x^3 + 670*x^2 + 245*x - 5, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 16 x^{18} + 42 x^{17} - 36 x^{16} - 28 x^{15} + 297 x^{14} - 2555 x^{13} + 6196 x^{12} + 10395 x^{11} - 40015 x^{10} + 13490 x^{9} + 93845 x^{8} - 94170 x^{7} - 74560 x^{6} + 101860 x^{5} - 6530 x^{4} - 13455 x^{3} + 670 x^{2} + 245 x - 5 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-5784566108091868330221282958984375=-\,5^{15}\cdot 11^{7}\cdot 9931^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.77$
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:  $5, 11, 9931$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5} a^{18} + \frac{1}{5} a^{17} + \frac{1}{5} a^{16} - \frac{1}{5} a^{15} - \frac{2}{5} a^{13} + \frac{1}{5} a^{12}$, $\frac{1}{679524878720912120948314371224398641605937835} a^{19} + \frac{31274772358623300128075697293128216220796262}{679524878720912120948314371224398641605937835} a^{18} + \frac{309477200037148371921854873046606906018596762}{679524878720912120948314371224398641605937835} a^{17} - \frac{39072536346309428787072597201787578481453546}{135904975744182424189662874244879728321187567} a^{16} + \frac{228793581370310725818171934808016712636514344}{679524878720912120948314371224398641605937835} a^{15} - \frac{331917207246097343432231762948667427596051287}{679524878720912120948314371224398641605937835} a^{14} + \frac{215457229187048674624160107056232260926536244}{679524878720912120948314371224398641605937835} a^{13} - \frac{278963329356203883806406747729682652961346959}{679524878720912120948314371224398641605937835} a^{12} + \frac{20320301586742746795323378660833002703675452}{135904975744182424189662874244879728321187567} a^{11} - \frac{25062174993570739002024143885648943363999482}{135904975744182424189662874244879728321187567} a^{10} + \frac{67930152739745946878077497425838073750817536}{135904975744182424189662874244879728321187567} a^{9} + \frac{52218002866321422792455224011183771444731655}{135904975744182424189662874244879728321187567} a^{8} - \frac{9292777867208541633079459295395543889426768}{135904975744182424189662874244879728321187567} a^{7} - \frac{26871270499365744882123231762759391116808727}{135904975744182424189662874244879728321187567} a^{6} + \frac{619650392240040648314706502289013677516715}{7152893460220127588929624960256827806378293} a^{5} + \frac{52283974489939564567662104762366276550559465}{135904975744182424189662874244879728321187567} a^{4} - \frac{12939477982202014404388299348383282050670897}{135904975744182424189662874244879728321187567} a^{3} - \frac{7016557347589837303650871296683683765717412}{135904975744182424189662874244879728321187567} a^{2} - \frac{60778764513591346714065000434380679257302657}{135904975744182424189662874244879728321187567} a - \frac{56612566496860314484203858160640653388606356}{135904975744182424189662874244879728321187567}$

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:  $16$
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:  \( 11626313713.8 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1023:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 324 conjugacy class representatives for t20n1023 are not computed
Character table for t20n1023 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.932312193828125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ $20$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.8.6.1$x^{8} + 143 x^{4} + 5929$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10}$
9931Data not computed