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Properties

Label 20.0.18055103407...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{44}\cdot 3^{16}\cdot 5^{22}$
Root discriminant $64.99$
Ramified primes $2, 3, 5$
Class number $2$ (GRH)
Class group $[2]$ (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![2259009, 0, -3547800, 0, 2727945, 0, -1310400, 0, 429690, 0, -102244, 0, 18470, 0, -2560, 0, 265, 0, -20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 20*x^18 + 265*x^16 - 2560*x^14 + 18470*x^12 - 102244*x^10 + 429690*x^8 - 1310400*x^6 + 2727945*x^4 - 3547800*x^2 + 2259009)
 
gp: K = bnfinit(x^20 - 20*x^18 + 265*x^16 - 2560*x^14 + 18470*x^12 - 102244*x^10 + 429690*x^8 - 1310400*x^6 + 2727945*x^4 - 3547800*x^2 + 2259009, 1)
 

Normalized defining polynomial

\( x^{20} - 20 x^{18} + 265 x^{16} - 2560 x^{14} + 18470 x^{12} - 102244 x^{10} + 429690 x^{8} - 1310400 x^{6} + 2727945 x^{4} - 3547800 x^{2} + 2259009 \)

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:  \(1805510340771840000000000000000000000=2^{44}\cdot 3^{16}\cdot 5^{22}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.99$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{3}$, $\frac{1}{66} a^{10} - \frac{7}{22} a^{8} - \frac{2}{33} a^{4} + \frac{7}{22} a^{2} - \frac{5}{22}$, $\frac{1}{66} a^{11} + \frac{1}{66} a^{9} - \frac{2}{33} a^{5} - \frac{1}{66} a^{3} - \frac{5}{22} a$, $\frac{1}{66} a^{12} + \frac{7}{22} a^{8} - \frac{2}{33} a^{6} + \frac{1}{22} a^{4} + \frac{5}{11} a^{2} + \frac{5}{22}$, $\frac{1}{66} a^{13} - \frac{1}{66} a^{9} - \frac{2}{33} a^{7} + \frac{1}{22} a^{5} - \frac{7}{33} a^{3} + \frac{5}{22} a$, $\frac{1}{66} a^{14} - \frac{25}{66} a^{8} + \frac{1}{22} a^{6} - \frac{3}{11} a^{4} - \frac{5}{11} a^{2} - \frac{5}{22}$, $\frac{1}{198} a^{15} + \frac{1}{198} a^{13} + \frac{1}{198} a^{11} - \frac{25}{198} a^{9} - \frac{1}{198} a^{7} - \frac{19}{198} a^{5} + \frac{29}{66} a^{3} - \frac{9}{22} a$, $\frac{1}{396} a^{16} - \frac{1}{198} a^{14} - \frac{1}{198} a^{12} + \frac{1}{198} a^{10} - \frac{1}{6} a^{9} + \frac{19}{198} a^{8} - \frac{4}{99} a^{6} + \frac{2}{33} a^{4} - \frac{1}{3} a^{3} - \frac{3}{11} a^{2} - \frac{1}{2} a - \frac{1}{44}$, $\frac{1}{396} a^{17} - \frac{1}{198} a^{11} - \frac{1}{22} a^{9} - \frac{1}{2} a^{8} - \frac{1}{22} a^{7} + \frac{5}{198} a^{5} + \frac{2}{11} a^{3} - \frac{9}{44} a - \frac{1}{2}$, $\frac{1}{899307992341836} a^{18} + \frac{107722000747}{899307992341836} a^{16} - \frac{3255245783011}{449653996170918} a^{14} + \frac{242873258497}{64236285167274} a^{12} + \frac{1008734634679}{449653996170918} a^{10} + \frac{41703730084211}{224826998085459} a^{8} - \frac{89088399535}{16653851710034} a^{6} - \frac{3365382098719}{49961555130102} a^{4} - \frac{10637816739197}{33307703420068} a^{2} + \frac{3149125642235}{33307703420068}$, $\frac{1}{150184434721086612} a^{19} + \frac{2325041182718}{3413282607297423} a^{17} + \frac{105751783591757}{75092217360543306} a^{15} + \frac{27859832496935}{5363729811467379} a^{13} - \frac{241354979107927}{37546108680271653} a^{11} + \frac{273212031019156}{3413282607297423} a^{9} - \frac{1}{2} a^{8} + \frac{393213454134899}{1390596617787839} a^{7} + \frac{94764018843047}{2781193235575678} a^{5} + \frac{30723237005347}{99923110260204} a^{3} + \frac{66517687423109}{252835748688698} a - \frac{1}{2}$

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:  \( 23657493051.91988 \) (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

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-2}, \sqrt{-5})\), 5.1.4050000.3, 10.2.167961600000000000.3, 10.0.33592320000000000.77, 10.0.5248800000000000.42

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
5Data not computed