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Properties

Label 20.0.14656213007...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 6029^{6}$
Root discriminant $45.53$
Ramified primes $5, 6029$
Class number $424$ (GRH)
Class group $[2, 212]$ (GRH)
Galois group 20T369

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![96739, 243078, 674906, 1044920, 996120, 803419, 530121, 225444, 131627, 8548, 22087, -8567, 6069, -1917, 1527, -277, 181, -45, 13, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 13*x^18 - 45*x^17 + 181*x^16 - 277*x^15 + 1527*x^14 - 1917*x^13 + 6069*x^12 - 8567*x^11 + 22087*x^10 + 8548*x^9 + 131627*x^8 + 225444*x^7 + 530121*x^6 + 803419*x^5 + 996120*x^4 + 1044920*x^3 + 674906*x^2 + 243078*x + 96739)
 
gp: K = bnfinit(x^20 - 3*x^19 + 13*x^18 - 45*x^17 + 181*x^16 - 277*x^15 + 1527*x^14 - 1917*x^13 + 6069*x^12 - 8567*x^11 + 22087*x^10 + 8548*x^9 + 131627*x^8 + 225444*x^7 + 530121*x^6 + 803419*x^5 + 996120*x^4 + 1044920*x^3 + 674906*x^2 + 243078*x + 96739, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 13 x^{18} - 45 x^{17} + 181 x^{16} - 277 x^{15} + 1527 x^{14} - 1917 x^{13} + 6069 x^{12} - 8567 x^{11} + 22087 x^{10} + 8548 x^{9} + 131627 x^{8} + 225444 x^{7} + 530121 x^{6} + 803419 x^{5} + 996120 x^{4} + 1044920 x^{3} + 674906 x^{2} + 243078 x + 96739 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1465621300755332525247833251953125=5^{15}\cdot 6029^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.53$
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:  $5, 6029$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is 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}{392825534163300480258852205971731849193375807517036001909} a^{19} - \frac{110602621885439021310407239044848030952486150610825528090}{392825534163300480258852205971731849193375807517036001909} a^{18} + \frac{82977953221965782372811889768202017074237921420441501673}{392825534163300480258852205971731849193375807517036001909} a^{17} - \frac{9364586468479207017098749864025514461674605245455625870}{23107384362547087074050129763043049952551518089237411877} a^{16} - \frac{72829498650470423831281973706147028699905524259640816062}{392825534163300480258852205971731849193375807517036001909} a^{15} - \frac{46381263465365463303319572913711084638931912638099354490}{392825534163300480258852205971731849193375807517036001909} a^{14} + \frac{125956125479510411298935023313696798335250494957596479472}{392825534163300480258852205971731849193375807517036001909} a^{13} - \frac{192417239323721183322231858866298563306833472491555271396}{392825534163300480258852205971731849193375807517036001909} a^{12} - \frac{64917961329219197354331846530754001162645185159309455447}{392825534163300480258852205971731849193375807517036001909} a^{11} + \frac{88080134207267401673946968246710525697844036553476115881}{392825534163300480258852205971731849193375807517036001909} a^{10} - \frac{98956393983773920048580714072436728096826201156457869606}{392825534163300480258852205971731849193375807517036001909} a^{9} - \frac{4473961595689267388116761723520524278171714787579090381}{392825534163300480258852205971731849193375807517036001909} a^{8} - \frac{145330792715476272587542567026770907161946853278859302896}{392825534163300480258852205971731849193375807517036001909} a^{7} + \frac{99237600884991325553541776231317657855374927705078178606}{392825534163300480258852205971731849193375807517036001909} a^{6} - \frac{150003832845430703087327121264620361979564954191501686139}{392825534163300480258852205971731849193375807517036001909} a^{5} + \frac{44029312803019099496444623847026330523892129180837333120}{392825534163300480258852205971731849193375807517036001909} a^{4} - \frac{84390510660952124011721164598759558418392008339884342284}{392825534163300480258852205971731849193375807517036001909} a^{3} - \frac{22881319924239390851568583418112625498495083100524438117}{392825534163300480258852205971731849193375807517036001909} a^{2} - \frac{158191141264498465588843648673771546242339458943978142314}{392825534163300480258852205971731849193375807517036001909} a + \frac{67827543605994755071393635558147641358406439283123803810}{392825534163300480258852205971731849193375807517036001909}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{212}$, which has order $424$ (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:  $9$
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:  \( 341439.528105 \) (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

\(\Q(\sqrt{5}) \), 5.5.753625.1, 10.10.2839753203125.1

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 $20$ $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
5Data not computed
6029Data not computed