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Properties

Label 20.20.1582635908...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{20}\cdot 5^{10}\cdot 11^{7}\cdot 28162171^{2}$
Root discriminant $57.54$
Ramified primes $2, 5, 11, 28162171$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T781

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-31, 470, -169, -16204, 42182, 65416, -202403, -134032, 308487, 137838, -204926, -65530, 68650, 14904, -12135, -1700, 1143, 94, -54, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 54*x^18 + 94*x^17 + 1143*x^16 - 1700*x^15 - 12135*x^14 + 14904*x^13 + 68650*x^12 - 65530*x^11 - 204926*x^10 + 137838*x^9 + 308487*x^8 - 134032*x^7 - 202403*x^6 + 65416*x^5 + 42182*x^4 - 16204*x^3 - 169*x^2 + 470*x - 31)
 
gp: K = bnfinit(x^20 - 2*x^19 - 54*x^18 + 94*x^17 + 1143*x^16 - 1700*x^15 - 12135*x^14 + 14904*x^13 + 68650*x^12 - 65530*x^11 - 204926*x^10 + 137838*x^9 + 308487*x^8 - 134032*x^7 - 202403*x^6 + 65416*x^5 + 42182*x^4 - 16204*x^3 - 169*x^2 + 470*x - 31, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 54 x^{18} + 94 x^{17} + 1143 x^{16} - 1700 x^{15} - 12135 x^{14} + 14904 x^{13} + 68650 x^{12} - 65530 x^{11} - 204926 x^{10} + 137838 x^{9} + 308487 x^{8} - 134032 x^{7} - 202403 x^{6} + 65416 x^{5} + 42182 x^{4} - 16204 x^{3} - 169 x^{2} + 470 x - 31 \)

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:  \(158263590809746088460400640000000000=2^{20}\cdot 5^{10}\cdot 11^{7}\cdot 28162171^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.54$
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:  $2, 5, 11, 28162171$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{88140118339651106758142066655693912090899} a^{19} + \frac{9202416435902738946439776671128296785553}{88140118339651106758142066655693912090899} a^{18} - \frac{372811792964503102706971857206454538542}{88140118339651106758142066655693912090899} a^{17} - \frac{29424495724079197782889883962534292335192}{88140118339651106758142066655693912090899} a^{16} - \frac{39305402514288688244805816204932255976521}{88140118339651106758142066655693912090899} a^{15} - \frac{38450992443249438501790347775817500416475}{88140118339651106758142066655693912090899} a^{14} + \frac{24171900260728527373532455269472997538816}{88140118339651106758142066655693912090899} a^{13} - \frac{3014720317633367941564348921601905364275}{88140118339651106758142066655693912090899} a^{12} - \frac{6511398152396280153216138641549945304524}{88140118339651106758142066655693912090899} a^{11} - \frac{13742122051435118449717759987265841716649}{88140118339651106758142066655693912090899} a^{10} - \frac{4927478476480531399210540867174922706362}{88140118339651106758142066655693912090899} a^{9} + \frac{22387357845228455615652681249677684836768}{88140118339651106758142066655693912090899} a^{8} - \frac{11118961226506631066775009978415875553067}{88140118339651106758142066655693912090899} a^{7} - \frac{1468747164821826193571819503134150674902}{88140118339651106758142066655693912090899} a^{6} - \frac{3796549049332870534574511790925475248210}{88140118339651106758142066655693912090899} a^{5} - \frac{12276156437939629921789844764681901307172}{88140118339651106758142066655693912090899} a^{4} + \frac{19638794979497071718580793620616397570699}{88140118339651106758142066655693912090899} a^{3} - \frac{13525073048301984754963495546973467568931}{88140118339651106758142066655693912090899} a^{2} - \frac{13380155136806392206213035351818904899443}{88140118339651106758142066655693912090899} a - \frac{27778639784481170375523686603074023839031}{88140118339651106758142066655693912090899}$

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:  \( 126774766446 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T781:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 115200
The 119 conjugacy class representatives for t20n781 are not computed
Character table for t20n781 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.4400.1, 10.10.968074628125.1

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 24 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 R ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
11.10.5.2$x^{10} + 1331 x^{4} - 14641 x^{2} + 805255$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
28162171Data not computed