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Properties

Label 20.0.36427451644...5625.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{12}\cdot 419^{4}\cdot 695771^{2}$
Root discriminant $33.73$
Ramified primes $5, 419, 695771$
Class number $40$ (GRH)
Class group $[2, 20]$ (GRH)
Galois group 20T1040

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5659, -7827, 13365, -27418, 31624, -30556, 31893, -22586, 16824, -11208, 6326, -3647, 1774, -771, 370, -132, 80, -11, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 9*x^18 - 11*x^17 + 80*x^16 - 132*x^15 + 370*x^14 - 771*x^13 + 1774*x^12 - 3647*x^11 + 6326*x^10 - 11208*x^9 + 16824*x^8 - 22586*x^7 + 31893*x^6 - 30556*x^5 + 31624*x^4 - 27418*x^3 + 13365*x^2 - 7827*x + 5659)
 
gp: K = bnfinit(x^20 + 9*x^18 - 11*x^17 + 80*x^16 - 132*x^15 + 370*x^14 - 771*x^13 + 1774*x^12 - 3647*x^11 + 6326*x^10 - 11208*x^9 + 16824*x^8 - 22586*x^7 + 31893*x^6 - 30556*x^5 + 31624*x^4 - 27418*x^3 + 13365*x^2 - 7827*x + 5659, 1)
 

Normalized defining polynomial

\( x^{20} + 9 x^{18} - 11 x^{17} + 80 x^{16} - 132 x^{15} + 370 x^{14} - 771 x^{13} + 1774 x^{12} - 3647 x^{11} + 6326 x^{10} - 11208 x^{9} + 16824 x^{8} - 22586 x^{7} + 31893 x^{6} - 30556 x^{5} + 31624 x^{4} - 27418 x^{3} + 13365 x^{2} - 7827 x + 5659 \)

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:  \(3642745164401140602041259765625=5^{12}\cdot 419^{4}\cdot 695771^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.73$
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:  $5, 419, 695771$
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}{41437015726543170955232796394283829196936247} a^{19} + \frac{13100680957691417932600850798361874789931937}{41437015726543170955232796394283829196936247} a^{18} - \frac{17815401637169020806084191420669276115819495}{41437015726543170955232796394283829196936247} a^{17} + \frac{3721194176992216772019395290790351927095297}{41437015726543170955232796394283829196936247} a^{16} - \frac{20522450038787192996747634997497392246984780}{41437015726543170955232796394283829196936247} a^{15} - \frac{20572709063093663019569911200248843692635082}{41437015726543170955232796394283829196936247} a^{14} + \frac{6065972532467330246514691981538125726335295}{41437015726543170955232796394283829196936247} a^{13} + \frac{2748606196100349432825168627883746334035987}{41437015726543170955232796394283829196936247} a^{12} - \frac{92343052733306320770190463391708358419970}{302459968806884459527246689009371016036031} a^{11} + \frac{20279980628717249479151200845872788708647666}{41437015726543170955232796394283829196936247} a^{10} + \frac{13336915102630653927586041851000356370743414}{41437015726543170955232796394283829196936247} a^{9} - \frac{8039028875840415169794664847186928136878014}{41437015726543170955232796394283829196936247} a^{8} + \frac{15159054528548777607895586798430250558971978}{41437015726543170955232796394283829196936247} a^{7} + \frac{5596881457572830251860813460770451040777898}{41437015726543170955232796394283829196936247} a^{6} + \frac{8611593540273022498605385190423737337311389}{41437015726543170955232796394283829196936247} a^{5} - \frac{2650774907425608596889676031368186825619802}{41437015726543170955232796394283829196936247} a^{4} - \frac{17262015741176363974967788833745642721201999}{41437015726543170955232796394283829196936247} a^{3} - \frac{4737117189226647075353180787319419717689510}{41437015726543170955232796394283829196936247} a^{2} + \frac{8676109026564742096381886852054176027004153}{41437015726543170955232796394283829196936247} a - \frac{7898937002570537250659625288375532950876634}{41437015726543170955232796394283829196936247}$

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

Class group and class number

$C_{2}\times C_{20}$, which has order $40$ (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:  \( 161078.71489 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1040:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 378 conjugacy class representatives for t20n1040 are not computed
Character table for t20n1040 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.911025153125.1

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 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
419Data not computed
695771Data not computed