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Properties

Label 20.4.16967521098...3125.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{15}\cdot 11^{18}$
Root discriminant $28.94$
Ramified primes $5, 11$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $C_2\times F_5$ (as 20T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![275, 275, 3025, 1375, -8140, 4345, 7095, -9460, 946, 5643, -4520, -150, 2051, -1080, -74, 320, -139, 5, 20, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 20*x^18 + 5*x^17 - 139*x^16 + 320*x^15 - 74*x^14 - 1080*x^13 + 2051*x^12 - 150*x^11 - 4520*x^10 + 5643*x^9 + 946*x^8 - 9460*x^7 + 7095*x^6 + 4345*x^5 - 8140*x^4 + 1375*x^3 + 3025*x^2 + 275*x + 275)
 
gp: K = bnfinit(x^20 - 8*x^19 + 20*x^18 + 5*x^17 - 139*x^16 + 320*x^15 - 74*x^14 - 1080*x^13 + 2051*x^12 - 150*x^11 - 4520*x^10 + 5643*x^9 + 946*x^8 - 9460*x^7 + 7095*x^6 + 4345*x^5 - 8140*x^4 + 1375*x^3 + 3025*x^2 + 275*x + 275, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 20 x^{18} + 5 x^{17} - 139 x^{16} + 320 x^{15} - 74 x^{14} - 1080 x^{13} + 2051 x^{12} - 150 x^{11} - 4520 x^{10} + 5643 x^{9} + 946 x^{8} - 9460 x^{7} + 7095 x^{6} + 4345 x^{5} - 8140 x^{4} + 1375 x^{3} + 3025 x^{2} + 275 x + 275 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(169675210983039290802001953125=5^{15}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.94$
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:  $5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{10} a^{12} + \frac{2}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{2} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{2} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{10} a^{13} + \frac{1}{10} a^{10} + \frac{2}{5} a^{9} + \frac{1}{5} a^{8} + \frac{3}{10} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{10} a^{4} - \frac{1}{2} a$, $\frac{1}{10} a^{14} + \frac{1}{10} a^{11} + \frac{2}{5} a^{10} + \frac{1}{5} a^{9} + \frac{3}{10} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{10} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{10} a^{15} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{2}$, $\frac{1}{10} a^{16} - \frac{2}{5} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{2} a$, $\frac{1}{10} a^{17} + \frac{2}{5} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{290} a^{18} + \frac{9}{290} a^{17} - \frac{7}{145} a^{16} - \frac{1}{145} a^{15} + \frac{9}{290} a^{14} + \frac{3}{145} a^{13} - \frac{1}{145} a^{12} + \frac{11}{58} a^{11} - \frac{31}{145} a^{10} - \frac{17}{145} a^{9} - \frac{1}{2} a^{8} + \frac{14}{29} a^{7} + \frac{68}{145} a^{6} + \frac{51}{290} a^{5} - \frac{64}{145} a^{4} + \frac{17}{58} a^{3} - \frac{13}{29} a^{2} + \frac{9}{29} a + \frac{1}{29}$, $\frac{1}{1522667992970003793291084950} a^{19} + \frac{2266977321878303145296507}{1522667992970003793291084950} a^{18} + \frac{6773990291204042030905639}{152266799297000379329108495} a^{17} + \frac{371648909846281366813322}{152266799297000379329108495} a^{16} - \frac{74432480434649783146548359}{1522667992970003793291084950} a^{15} + \frac{5031632950997647515874102}{152266799297000379329108495} a^{14} - \frac{21833179443902556562042887}{761333996485001896645542475} a^{13} + \frac{5796575703779273149222423}{304533598594000758658216990} a^{12} + \frac{211896652244235516789051143}{761333996485001896645542475} a^{11} - \frac{5094983111935032768689669}{30453359859400075865821699} a^{10} + \frac{11130162058058357900779263}{304533598594000758658216990} a^{9} + \frac{294113891802637106805338274}{761333996485001896645542475} a^{8} + \frac{279775467408119164897674773}{761333996485001896645542475} a^{7} + \frac{50588424081369379389024501}{304533598594000758658216990} a^{6} + \frac{15997386538840018913567638}{152266799297000379329108495} a^{5} + \frac{4179558817624777932002997}{60906719718800151731643398} a^{4} + \frac{19366951186871819718576203}{152266799297000379329108495} a^{3} - \frac{14879272604886868037021350}{30453359859400075865821699} a^{2} + \frac{14145830017507402057408876}{30453359859400075865821699} a - \frac{10875117228237674576402557}{30453359859400075865821699}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  $11$
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:  \( 3624334.381450157 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T9):

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

\(\Q(\sqrt{5}) \), 4.4.15125.1, 5.1.1830125.1, 10.2.16746787578125.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$11$11.10.9.8$x^{10} + 33$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.8$x^{10} + 33$$10$$1$$9$$C_{10}$$[\ ]_{10}$