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Properties

Label 20.20.1807841593...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{28}\cdot 5^{22}\cdot 7^{10}$
Root discriminant $41.01$
Ramified primes $2, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times F_5$ (as 20T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 80, -780, 760, 6280, -9120, -19000, 24400, 29300, -28100, -25376, 16200, 12770, -4780, -3690, 660, 570, -30, -40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 40*x^18 - 30*x^17 + 570*x^16 + 660*x^15 - 3690*x^14 - 4780*x^13 + 12770*x^12 + 16200*x^11 - 25376*x^10 - 28100*x^9 + 29300*x^8 + 24400*x^7 - 19000*x^6 - 9120*x^5 + 6280*x^4 + 760*x^3 - 780*x^2 + 80*x + 4)
 
gp: K = bnfinit(x^20 - 40*x^18 - 30*x^17 + 570*x^16 + 660*x^15 - 3690*x^14 - 4780*x^13 + 12770*x^12 + 16200*x^11 - 25376*x^10 - 28100*x^9 + 29300*x^8 + 24400*x^7 - 19000*x^6 - 9120*x^5 + 6280*x^4 + 760*x^3 - 780*x^2 + 80*x + 4, 1)
 

Normalized defining polynomial

\( x^{20} - 40 x^{18} - 30 x^{17} + 570 x^{16} + 660 x^{15} - 3690 x^{14} - 4780 x^{13} + 12770 x^{12} + 16200 x^{11} - 25376 x^{10} - 28100 x^{9} + 29300 x^{8} + 24400 x^{7} - 19000 x^{6} - 9120 x^{5} + 6280 x^{4} + 760 x^{3} - 780 x^{2} + 80 x + 4 \)

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:  \(180784159360000000000000000000000=2^{28}\cdot 5^{22}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.01$
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:  $2, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{2} a^{14}$, $\frac{1}{2} a^{15}$, $\frac{1}{2} a^{16}$, $\frac{1}{2} a^{17}$, $\frac{1}{348} a^{18} - \frac{13}{87} a^{17} + \frac{19}{87} a^{16} - \frac{20}{87} a^{15} - \frac{3}{29} a^{14} + \frac{19}{87} a^{13} - \frac{41}{174} a^{12} - \frac{31}{174} a^{11} - \frac{13}{58} a^{10} + \frac{25}{87} a^{9} - \frac{17}{58} a^{8} - \frac{16}{87} a^{7} + \frac{9}{29} a^{6} - \frac{2}{29} a^{5} - \frac{16}{87} a^{4} - \frac{14}{87} a^{3} + \frac{32}{87} a^{2} - \frac{14}{29} a - \frac{4}{87}$, $\frac{1}{845334585316617996} a^{19} - \frac{273474190022917}{845334585316617996} a^{18} + \frac{26077381151053516}{211333646329154499} a^{17} + \frac{824317748144396}{7287367114798431} a^{16} - \frac{30772524315150317}{140889097552769666} a^{15} - \frac{10258377116777953}{422667292658308998} a^{14} - \frac{19348939530037865}{422667292658308998} a^{13} - \frac{42531750559562261}{211333646329154499} a^{12} - \frac{5159262331343922}{70444548776384833} a^{11} - \frac{35905478464577791}{422667292658308998} a^{10} - \frac{50756736782487051}{140889097552769666} a^{9} + \frac{156367079043516097}{422667292658308998} a^{8} + \frac{2717032755110053}{70444548776384833} a^{7} + \frac{22211665689822061}{70444548776384833} a^{6} - \frac{11133705200355247}{211333646329154499} a^{5} + \frac{55574508164167417}{211333646329154499} a^{4} + \frac{65094765387499967}{211333646329154499} a^{3} - \frac{6480948048512046}{70444548776384833} a^{2} - \frac{57656464573207228}{211333646329154499} a + \frac{784023966847350}{70444548776384833}$

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:  \( 6884203494.14 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{35}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{5}, \sqrt{7})\), 5.5.2450000.1, 10.10.30012500000000.1, 10.10.13445600000000000.1, 10.10.2689120000000000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 sibling: 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
5Data not computed
$7$7.4.2.1$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}$
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}$