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Properties

Label 20.0.96479729228...3024.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{63}\cdot 3^{21}$
Root discriminant $28.13$
Ramified primes $2, 3$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T547

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![564, 1464, 1128, -60, 2097, 1104, -1008, 1848, -294, -456, 1416, -1200, 748, -104, -136, 160, -62, 8, 8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 8*x^18 + 8*x^17 - 62*x^16 + 160*x^15 - 136*x^14 - 104*x^13 + 748*x^12 - 1200*x^11 + 1416*x^10 - 456*x^9 - 294*x^8 + 1848*x^7 - 1008*x^6 + 1104*x^5 + 2097*x^4 - 60*x^3 + 1128*x^2 + 1464*x + 564)
 
gp: K = bnfinit(x^20 - 4*x^19 + 8*x^18 + 8*x^17 - 62*x^16 + 160*x^15 - 136*x^14 - 104*x^13 + 748*x^12 - 1200*x^11 + 1416*x^10 - 456*x^9 - 294*x^8 + 1848*x^7 - 1008*x^6 + 1104*x^5 + 2097*x^4 - 60*x^3 + 1128*x^2 + 1464*x + 564, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 8 x^{18} + 8 x^{17} - 62 x^{16} + 160 x^{15} - 136 x^{14} - 104 x^{13} + 748 x^{12} - 1200 x^{11} + 1416 x^{10} - 456 x^{9} - 294 x^{8} + 1848 x^{7} - 1008 x^{6} + 1104 x^{5} + 2097 x^{4} - 60 x^{3} + 1128 x^{2} + 1464 x + 564 \)

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:  \(96479729228174488169059713024=2^{63}\cdot 3^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.13$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{4} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{6} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{36246703791470795427093963766300} a^{19} + \frac{47647051568985058944435936323}{1394103991979645977965152452550} a^{18} + \frac{4722332153887808434003702637527}{18123351895735397713546981883150} a^{17} - \frac{2210537046371818267950434807846}{9061675947867698856773490941575} a^{16} - \frac{1766694234517120370352590334893}{3624670379147079542709396376630} a^{15} - \frac{118141415946023324412518786756}{362467037914707954270939637663} a^{14} - \frac{2873872447670782207505147442334}{9061675947867698856773490941575} a^{13} - \frac{2158902090506453438875981834044}{9061675947867698856773490941575} a^{12} + \frac{4228677457383311709250898464949}{9061675947867698856773490941575} a^{11} + \frac{2944619323062802104625332211598}{9061675947867698856773490941575} a^{10} + \frac{110048915992357873616318534435}{362467037914707954270939637663} a^{9} + \frac{255440142494862707369287504247}{697051995989822988982576226275} a^{8} - \frac{8514147083772028776063753990253}{18123351895735397713546981883150} a^{7} - \frac{1070716469204115970996679183291}{9061675947867698856773490941575} a^{6} + \frac{1927260169639966132049880993741}{9061675947867698856773490941575} a^{5} - \frac{4253631076017341955315924174442}{9061675947867698856773490941575} a^{4} - \frac{14374038393907665909714226373039}{36246703791470795427093963766300} a^{3} + \frac{8710407328947429388658922543231}{18123351895735397713546981883150} a^{2} - \frac{7108668940866177927383376460099}{18123351895735397713546981883150} a - \frac{3521887360479636518705421162108}{9061675947867698856773490941575}$

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:  $9$
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:  \( 5069328.62837 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T547:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 28800
The 41 conjugacy class representatives for t20n547
Character table for t20n547 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.0.13824.1, 10.2.5283615080448.1

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 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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.12.15.3$x^{12} - 3 x^{11} - 3 x^{9} - 3 x^{8} + 3 x^{7} - 3 x^{6} - 3 x^{4} - 3$$12$$1$$15$12T42$[3/2]_{4}^{6}$