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Properties

Label 20.0.51563828207...2241.1
Degree $20$
Signature $[0, 10]$
Discriminant $7^{10}\cdot 23^{2}\cdot 431^{4}$
Root discriminant $12.18$
Ramified primes $7, 23, 431$
Class number $1$
Class group Trivial
Galois group 20T368

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 8, -19, 47, -82, 133, -205, 303, -377, 422, -418, 385, -318, 237, -150, 87, -41, 17, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 17*x^18 - 41*x^17 + 87*x^16 - 150*x^15 + 237*x^14 - 318*x^13 + 385*x^12 - 418*x^11 + 422*x^10 - 377*x^9 + 303*x^8 - 205*x^7 + 133*x^6 - 82*x^5 + 47*x^4 - 19*x^3 + 8*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^20 - 5*x^19 + 17*x^18 - 41*x^17 + 87*x^16 - 150*x^15 + 237*x^14 - 318*x^13 + 385*x^12 - 418*x^11 + 422*x^10 - 377*x^9 + 303*x^8 - 205*x^7 + 133*x^6 - 82*x^5 + 47*x^4 - 19*x^3 + 8*x^2 - 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 17 x^{18} - 41 x^{17} + 87 x^{16} - 150 x^{15} + 237 x^{14} - 318 x^{13} + 385 x^{12} - 418 x^{11} + 422 x^{10} - 377 x^{9} + 303 x^{8} - 205 x^{7} + 133 x^{6} - 82 x^{5} + 47 x^{4} - 19 x^{3} + 8 x^{2} - 3 x + 1 \)

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:  \(5156382820784106642241=7^{10}\cdot 23^{2}\cdot 431^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.18$
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:  $7, 23, 431$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{17} - \frac{1}{10} a^{16} + \frac{1}{10} a^{15} + \frac{1}{5} a^{14} + \frac{1}{10} a^{13} + \frac{1}{5} a^{12} - \frac{3}{10} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{3}{10} a^{5} - \frac{1}{10} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a - \frac{3}{10}$, $\frac{1}{350} a^{18} - \frac{9}{350} a^{17} + \frac{7}{50} a^{16} - \frac{13}{175} a^{15} - \frac{1}{70} a^{14} - \frac{13}{175} a^{13} + \frac{11}{350} a^{12} + \frac{46}{175} a^{11} + \frac{74}{175} a^{10} + \frac{11}{175} a^{9} + \frac{46}{175} a^{8} + \frac{3}{25} a^{7} + \frac{117}{350} a^{6} + \frac{17}{175} a^{5} - \frac{121}{350} a^{4} - \frac{6}{25} a^{3} - \frac{4}{175} a^{2} - \frac{1}{350} a + \frac{22}{175}$, $\frac{1}{1247884750} a^{19} + \frac{38742}{89134625} a^{18} + \frac{30509463}{623942375} a^{17} + \frac{106677901}{623942375} a^{16} - \frac{136571201}{623942375} a^{15} + \frac{472075939}{1247884750} a^{14} - \frac{145650368}{623942375} a^{13} + \frac{224605942}{623942375} a^{12} - \frac{365615653}{1247884750} a^{11} + \frac{179478339}{623942375} a^{10} + \frac{485967401}{1247884750} a^{9} + \frac{180528783}{623942375} a^{8} - \frac{162086142}{623942375} a^{7} + \frac{180254783}{1247884750} a^{6} + \frac{203426851}{623942375} a^{5} + \frac{79130252}{623942375} a^{4} + \frac{142940069}{1247884750} a^{3} + \frac{235577773}{1247884750} a^{2} - \frac{3662302}{89134625} a + \frac{143805734}{623942375}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 269.656794871 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T368:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7680
The 72 conjugacy class representatives for t20n368 are not computed
Character table for t20n368 is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), 5.1.3017.1, 10.0.3122085127.1, 10.0.209352647.1, 10.2.71807957921.1

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.12.6.1$x^{12} + 294 x^{8} + 3430 x^{6} + 21609 x^{4} + 487403 x^{2} + 2941225$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
431Data not computed