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Properties

Label 20.20.4270049530...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{28}\cdot 3^{10}\cdot 5^{20}\cdot 7^{10}$
Root discriminant $60.47$
Ramified primes $2, 3, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2\times F_5$ (as 20T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3008, 21120, -22400, -106560, 182400, 145056, -327840, -86400, 273120, 26400, -128056, -4320, 36400, 360, -6400, -12, 680, 0, -40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 40*x^18 + 680*x^16 - 12*x^15 - 6400*x^14 + 360*x^13 + 36400*x^12 - 4320*x^11 - 128056*x^10 + 26400*x^9 + 273120*x^8 - 86400*x^7 - 327840*x^6 + 145056*x^5 + 182400*x^4 - 106560*x^3 - 22400*x^2 + 21120*x - 3008)
 
gp: K = bnfinit(x^20 - 40*x^18 + 680*x^16 - 12*x^15 - 6400*x^14 + 360*x^13 + 36400*x^12 - 4320*x^11 - 128056*x^10 + 26400*x^9 + 273120*x^8 - 86400*x^7 - 327840*x^6 + 145056*x^5 + 182400*x^4 - 106560*x^3 - 22400*x^2 + 21120*x - 3008, 1)
 

Normalized defining polynomial

\( x^{20} - 40 x^{18} + 680 x^{16} - 12 x^{15} - 6400 x^{14} + 360 x^{13} + 36400 x^{12} - 4320 x^{11} - 128056 x^{10} + 26400 x^{9} + 273120 x^{8} - 86400 x^{7} - 327840 x^{6} + 145056 x^{5} + 182400 x^{4} - 106560 x^{3} - 22400 x^{2} + 21120 x - 3008 \)

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:  \(427004953041945600000000000000000000=2^{28}\cdot 3^{10}\cdot 5^{20}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.47$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{4} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{8} a^{10}$, $\frac{1}{8} a^{11}$, $\frac{1}{8} a^{12}$, $\frac{1}{8} a^{13}$, $\frac{1}{16} a^{14}$, $\frac{1}{80} a^{15} - \frac{1}{5} a^{5} - \frac{1}{5}$, $\frac{1}{80} a^{16} - \frac{1}{5} a^{6} - \frac{1}{5} a$, $\frac{1}{160} a^{17} - \frac{1}{10} a^{7} + \frac{2}{5} a^{2}$, $\frac{1}{7520} a^{18} + \frac{7}{7520} a^{17} - \frac{9}{1880} a^{16} + \frac{11}{1880} a^{15} + \frac{7}{752} a^{14} - \frac{7}{188} a^{13} + \frac{3}{94} a^{12} - \frac{5}{94} a^{11} + \frac{3}{94} a^{10} + \frac{2}{47} a^{9} + \frac{39}{470} a^{8} - \frac{69}{940} a^{7} - \frac{42}{235} a^{6} + \frac{3}{235} a^{5} - \frac{3}{47} a^{4} - \frac{78}{235} a^{3} - \frac{46}{235} a^{2} + \frac{93}{235} a - \frac{1}{5}$, $\frac{1}{7520} a^{19} + \frac{9}{7520} a^{17} + \frac{7}{3760} a^{16} + \frac{11}{1880} a^{15} + \frac{17}{752} a^{14} + \frac{2}{47} a^{13} - \frac{5}{188} a^{12} + \frac{11}{376} a^{11} - \frac{21}{376} a^{10} + \frac{33}{940} a^{9} + \frac{9}{94} a^{8} - \frac{27}{235} a^{7} - \frac{32}{235} a^{6} + \frac{58}{235} a^{5} + \frac{27}{235} a^{4} + \frac{6}{47} a^{3} - \frac{102}{235} a^{2} - \frac{87}{235} a - \frac{1}{5}$

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:  \( 622976098097 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times F_5$ (as 20T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_2^2\times F_5$
Character table for $C_2^2\times F_5$

Intermediate fields

\(\Q(\sqrt{21}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{3}, \sqrt{7})\), 5.5.2450000.1, 10.10.10210252500000000.1, 10.10.93350880000000000.1, 10.10.2689120000000000.1

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 R R R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.10.10.10$x^{10} + 10 x^{8} + 5 x^{6} + 10 x^{5} - 20 x^{4} - 20 x^{2} + 2$$5$$2$$10$$F_{5}\times C_2$$[5/4]_{4}^{2}$
5.10.10.10$x^{10} + 10 x^{8} + 5 x^{6} + 10 x^{5} - 20 x^{4} - 20 x^{2} + 2$$5$$2$$10$$F_{5}\times C_2$$[5/4]_{4}^{2}$
$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}$