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Properties

Label 20.0.64739954558...9344.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{15}\cdot 7^{8}\cdot 17^{11}$
Root discriminant $17.40$
Ramified primes $2, 7, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T525

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19, 2, 89, -58, 114, -119, 243, -181, 133, -132, 118, -75, 1, 45, -38, -13, 46, -41, 22, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 22*x^18 - 41*x^17 + 46*x^16 - 13*x^15 - 38*x^14 + 45*x^13 + x^12 - 75*x^11 + 118*x^10 - 132*x^9 + 133*x^8 - 181*x^7 + 243*x^6 - 119*x^5 + 114*x^4 - 58*x^3 + 89*x^2 + 2*x + 19)
 
gp: K = bnfinit(x^20 - 7*x^19 + 22*x^18 - 41*x^17 + 46*x^16 - 13*x^15 - 38*x^14 + 45*x^13 + x^12 - 75*x^11 + 118*x^10 - 132*x^9 + 133*x^8 - 181*x^7 + 243*x^6 - 119*x^5 + 114*x^4 - 58*x^3 + 89*x^2 + 2*x + 19, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 22 x^{18} - 41 x^{17} + 46 x^{16} - 13 x^{15} - 38 x^{14} + 45 x^{13} + x^{12} - 75 x^{11} + 118 x^{10} - 132 x^{9} + 133 x^{8} - 181 x^{7} + 243 x^{6} - 119 x^{5} + 114 x^{4} - 58 x^{3} + 89 x^{2} + 2 x + 19 \)

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:  \(6473995455893963605049344=2^{15}\cdot 7^{8}\cdot 17^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.40$
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:  $2, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{109689082071964220298577} a^{19} + \frac{14983137063540600413167}{109689082071964220298577} a^{18} - \frac{28642712497047487197174}{109689082071964220298577} a^{17} - \frac{12624883293507031450682}{109689082071964220298577} a^{16} - \frac{15649494876923491685990}{109689082071964220298577} a^{15} + \frac{2703836674426987692551}{5773109582734958963083} a^{14} + \frac{2511449149372963676872}{5773109582734958963083} a^{13} - \frac{13017195102551292017077}{109689082071964220298577} a^{12} + \frac{282220350323942336118}{5773109582734958963083} a^{11} - \frac{9733298391236538028255}{109689082071964220298577} a^{10} - \frac{5529988022869065617868}{109689082071964220298577} a^{9} + \frac{32095051562980484380242}{109689082071964220298577} a^{8} + \frac{52923581664116815429123}{109689082071964220298577} a^{7} + \frac{7394090948493475852622}{109689082071964220298577} a^{6} - \frac{44016933131759501262916}{109689082071964220298577} a^{5} - \frac{285039548589609206368}{4769090524868009578199} a^{4} - \frac{345503870798108363021}{8437621697843401561429} a^{3} + \frac{861031253471101898109}{4769090524868009578199} a^{2} + \frac{46190144766769548083297}{109689082071964220298577} a - \frac{2593194290012540599009}{5773109582734958963083}$

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:  $9$
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:  \( 13469.2003837 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T525:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20480
The 152 conjugacy class representatives for t20n525 are not computed
Character table for t20n525 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 5.1.14161.1, 10.2.3409076657.1

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ $20$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ $20$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.10.15.7$x^{10} - 10 x^{8} + 56 x^{6} - 176 x^{4} + 272 x^{2} - 1184$$2$$5$$15$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 3]^{5}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$