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Properties

Label 20.20.9088634776...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{16}\cdot 5^{13}\cdot 7^{6}\cdot 23^{4}\cdot 431^{4}$
Root discriminant $55.97$
Ramified primes $2, 5, 7, 23, 431$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T369

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, 290, 1481, -2945, -14791, 19129, 49465, -68783, -56208, 100553, 17649, -64498, 4258, 19885, -3226, -3005, 590, 206, -42, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 42*x^18 + 206*x^17 + 590*x^16 - 3005*x^15 - 3226*x^14 + 19885*x^13 + 4258*x^12 - 64498*x^11 + 17649*x^10 + 100553*x^9 - 56208*x^8 - 68783*x^7 + 49465*x^6 + 19129*x^5 - 14791*x^4 - 2945*x^3 + 1481*x^2 + 290*x - 5)
 
gp: K = bnfinit(x^20 - 5*x^19 - 42*x^18 + 206*x^17 + 590*x^16 - 3005*x^15 - 3226*x^14 + 19885*x^13 + 4258*x^12 - 64498*x^11 + 17649*x^10 + 100553*x^9 - 56208*x^8 - 68783*x^7 + 49465*x^6 + 19129*x^5 - 14791*x^4 - 2945*x^3 + 1481*x^2 + 290*x - 5, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 42 x^{18} + 206 x^{17} + 590 x^{16} - 3005 x^{15} - 3226 x^{14} + 19885 x^{13} + 4258 x^{12} - 64498 x^{11} + 17649 x^{10} + 100553 x^{9} - 56208 x^{8} - 68783 x^{7} + 49465 x^{6} + 19129 x^{5} - 14791 x^{4} - 2945 x^{3} + 1481 x^{2} + 290 x - 5 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(90886347761592416951120000000000000=2^{16}\cdot 5^{13}\cdot 7^{6}\cdot 23^{4}\cdot 431^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.97$
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:  $2, 5, 7, 23, 431$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{25} a^{18} + \frac{1}{25} a^{17} - \frac{8}{25} a^{15} - \frac{8}{25} a^{14} + \frac{9}{25} a^{13} - \frac{2}{5} a^{12} - \frac{1}{25} a^{11} - \frac{8}{25} a^{10} - \frac{7}{25} a^{9} - \frac{6}{25} a^{8} - \frac{2}{5} a^{7} - \frac{9}{25} a^{6} + \frac{3}{25} a^{5} + \frac{9}{25} a^{4} - \frac{9}{25} a^{3} + \frac{4}{25} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{2250541316086906727549628137601775} a^{19} - \frac{33226707432380626546752403315409}{2250541316086906727549628137601775} a^{18} + \frac{218482442780726733405788233326903}{450108263217381345509925627520355} a^{17} - \frac{933190985915124084397669252733833}{2250541316086906727549628137601775} a^{16} + \frac{2266649880481156716377629282983}{20647168037494557133482826950475} a^{15} + \frac{456714024882468561653555238959939}{2250541316086906727549628137601775} a^{14} + \frac{40973768852817772039172343274364}{90021652643476269101985125504071} a^{13} + \frac{1059000174198913209875368034399849}{2250541316086906727549628137601775} a^{12} + \frac{568015334158473425551144766662227}{2250541316086906727549628137601775} a^{11} + \frac{979553413043769662547540561943823}{2250541316086906727549628137601775} a^{10} + \frac{463903714797877833208760121837114}{2250541316086906727549628137601775} a^{9} - \frac{36157234071080618995174281051966}{90021652643476269101985125504071} a^{8} - \frac{884165558848601067743198437200709}{2250541316086906727549628137601775} a^{7} + \frac{299020714117937358750042436077768}{2250541316086906727549628137601775} a^{6} + \frac{227245056307836287631148801116254}{2250541316086906727549628137601775} a^{5} + \frac{57671058613517260026839344749151}{2250541316086906727549628137601775} a^{4} + \frac{57051654746020638617511134751294}{2250541316086906727549628137601775} a^{3} - \frac{55169893141823090759843233163147}{450108263217381345509925627520355} a^{2} + \frac{140934927969856569596774248432816}{450108263217381345509925627520355} a + \frac{42658206856594826254015734760878}{90021652643476269101985125504071}$

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:  $19$
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:  \( 306744996583 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T369:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7680
The 72 conjugacy class representatives for t20n369 are not computed
Character table for t20n369 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.396520.1, 10.10.19653513800000.1

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 R $20$ R R ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.12.0.1$x^{12} - 26 x^{10} + 275 x^{8} - 1500 x^{6} + 4375 x^{4} - 6250 x^{2} + 7221$$1$$12$$0$$C_{12}$$[\ ]^{12}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.12.6.2$x^{12} + 7203 x^{4} - 16807 x^{2} + 588245$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
$23$23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
431Data not computed