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Properties

Label 20.16.1818109779...0000.2
Degree $20$
Signature $[16, 2]$
Discriminant $2^{24}\cdot 5^{10}\cdot 13^{4}\cdot 29^{10}\cdot 31^{4}$
Root discriminant $91.83$
Ramified primes $2, 5, 13, 29, 31$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T1028

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2533049, -11874188, 73196147, -5953928, -150272139, -19969942, 61562848, 13834086, -3948130, -1532790, -2216342, -530826, 529618, 140894, -52856, -12350, 2899, 458, -85, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 85*x^18 + 458*x^17 + 2899*x^16 - 12350*x^15 - 52856*x^14 + 140894*x^13 + 529618*x^12 - 530826*x^11 - 2216342*x^10 - 1532790*x^9 - 3948130*x^8 + 13834086*x^7 + 61562848*x^6 - 19969942*x^5 - 150272139*x^4 - 5953928*x^3 + 73196147*x^2 - 11874188*x - 2533049)
 
gp: K = bnfinit(x^20 - 6*x^19 - 85*x^18 + 458*x^17 + 2899*x^16 - 12350*x^15 - 52856*x^14 + 140894*x^13 + 529618*x^12 - 530826*x^11 - 2216342*x^10 - 1532790*x^9 - 3948130*x^8 + 13834086*x^7 + 61562848*x^6 - 19969942*x^5 - 150272139*x^4 - 5953928*x^3 + 73196147*x^2 - 11874188*x - 2533049, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} - 85 x^{18} + 458 x^{17} + 2899 x^{16} - 12350 x^{15} - 52856 x^{14} + 140894 x^{13} + 529618 x^{12} - 530826 x^{11} - 2216342 x^{10} - 1532790 x^{9} - 3948130 x^{8} + 13834086 x^{7} + 61562848 x^{6} - 19969942 x^{5} - 150272139 x^{4} - 5953928 x^{3} + 73196147 x^{2} - 11874188 x - 2533049 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1818109779441283591477572567040000000000=2^{24}\cdot 5^{10}\cdot 13^{4}\cdot 29^{10}\cdot 31^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.83$
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:  $2, 5, 13, 29, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} + \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{14} - \frac{1}{4} a^{10} + \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{8} a - \frac{3}{8}$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{14} - \frac{1}{4} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{8} + \frac{1}{8} a^{6} + \frac{1}{4} a^{3} + \frac{1}{8} a^{2} + \frac{1}{8}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{14} - \frac{1}{8} a^{11} - \frac{1}{8} a^{9} + \frac{1}{8} a^{8} + \frac{1}{8} a^{6} - \frac{1}{2} a^{5} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2} + \frac{3}{8}$, $\frac{1}{8} a^{18} - \frac{1}{8} a^{14} - \frac{1}{8} a^{12} + \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{8} a^{8} + \frac{1}{8} a^{6} - \frac{1}{2} a^{5} - \frac{3}{8} a^{4} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{3}{8}$, $\frac{1}{289232958680788799870702027315153491799063169973271505840874088716609264} a^{19} - \frac{30633106744881006604401882731158501195334945890821739676344644131499}{9330095441315767737764581526295273929002037741073274381963680281180944} a^{18} + \frac{2107487542528439111538045265414538460552552864726992354298558010031625}{36154119835098599983837753414394186474882896246658938230109261089576158} a^{17} + \frac{2757656473144902977442852209666271817900632583632685880572569085003349}{72308239670197199967675506828788372949765792493317876460218522179152316} a^{16} - \frac{8187759949446447304798513643050025877496688206154993161170491570441601}{289232958680788799870702027315153491799063169973271505840874088716609264} a^{15} - \frac{11175644970261107161536721115377109483300281366480194298437559681570057}{289232958680788799870702027315153491799063169973271505840874088716609264} a^{14} - \frac{32291270308524863431404990440682346874770630631344785343876324688584419}{289232958680788799870702027315153491799063169973271505840874088716609264} a^{13} + \frac{12726075885847408535978352532650497157379847306317038045436897592667183}{289232958680788799870702027315153491799063169973271505840874088716609264} a^{12} - \frac{41456072430224737057952136616186138571041474161040489909058163496570745}{289232958680788799870702027315153491799063169973271505840874088716609264} a^{11} - \frac{10336356040814155047947513993479577913990781510553309824077884181696873}{289232958680788799870702027315153491799063169973271505840874088716609264} a^{10} - \frac{72304733451637375341286277471174968222410351974897718139744700424960961}{289232958680788799870702027315153491799063169973271505840874088716609264} a^{9} + \frac{35912884052763036771172707123346388292911053462001123862626080128801373}{289232958680788799870702027315153491799063169973271505840874088716609264} a^{8} + \frac{14466108291616125704313420109458405113433691223762588370915243604640573}{289232958680788799870702027315153491799063169973271505840874088716609264} a^{7} + \frac{1567610440504106161626691096909864258112463803822831833630383040245395}{9330095441315767737764581526295273929002037741073274381963680281180944} a^{6} + \frac{121410853100229034143991456698775722016824729682083349831676456859071103}{289232958680788799870702027315153491799063169973271505840874088716609264} a^{5} + \frac{73085216952699552650585500960904368313793755591665316156673984610500525}{289232958680788799870702027315153491799063169973271505840874088716609264} a^{4} + \frac{3167624217253261736632730353385051952154860761339593844410459511962763}{72308239670197199967675506828788372949765792493317876460218522179152316} a^{3} - \frac{34191724339526217559994577384101416793169394145440988236861881269155}{144616479340394399935351013657576745899531584986635752920437044358304632} a^{2} + \frac{62235044498538468285156376673842166643019531469810214120329057830240141}{289232958680788799870702027315153491799063169973271505840874088716609264} a - \frac{55466390075121069998877769883130188075693019919215888694417735468430733}{289232958680788799870702027315153491799063169973271505840874088716609264}$

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:  $17$
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:  \( 10820789941400 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1028:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 228 conjugacy class representatives for t20n1028 are not computed
Character table for t20n1028 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.109268775200000.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
$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}$
5.8.4.1$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}$
$13$13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.6.5.2$x^{6} + 58$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
29.10.5.1$x^{10} - 1682 x^{6} + 707281 x^{2} - 2481849029$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$31$31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.6.4.3$x^{6} + 713 x^{3} + 138384$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$