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Properties

Label 20.0.76594611910...4224.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 61^{10}$
Root discriminant $22.09$
Ramified primes $2, 61$
Class number $1$
Class group Trivial
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1600, -5440, 4816, 3888, -5308, -2152, 3214, 168, -1375, 352, 140, -148, 242, 160, -42, -28, 20, -4, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^18 - 4*x^17 + 20*x^16 - 28*x^15 - 42*x^14 + 160*x^13 + 242*x^12 - 148*x^11 + 140*x^10 + 352*x^9 - 1375*x^8 + 168*x^7 + 3214*x^6 - 2152*x^5 - 5308*x^4 + 3888*x^3 + 4816*x^2 - 5440*x + 1600)
 
gp: K = bnfinit(x^20 - 4*x^18 - 4*x^17 + 20*x^16 - 28*x^15 - 42*x^14 + 160*x^13 + 242*x^12 - 148*x^11 + 140*x^10 + 352*x^9 - 1375*x^8 + 168*x^7 + 3214*x^6 - 2152*x^5 - 5308*x^4 + 3888*x^3 + 4816*x^2 - 5440*x + 1600, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{18} - 4 x^{17} + 20 x^{16} - 28 x^{15} - 42 x^{14} + 160 x^{13} + 242 x^{12} - 148 x^{11} + 140 x^{10} + 352 x^{9} - 1375 x^{8} + 168 x^{7} + 3214 x^{6} - 2152 x^{5} - 5308 x^{4} + 3888 x^{3} + 4816 x^{2} - 5440 x + 1600 \)

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:  \(765946119106374437095604224=2^{30}\cdot 61^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.09$
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:  $2, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7}$, $\frac{1}{10} a^{14} - \frac{1}{5} a^{13} - \frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{1}{10} a^{9} - \frac{2}{5} a^{6} + \frac{1}{5} a^{4} - \frac{3}{10} a^{3} - \frac{3}{10} a^{2} + \frac{2}{5} a$, $\frac{1}{10} a^{15} - \frac{1}{10} a^{13} - \frac{1}{10} a^{12} + \frac{1}{10} a^{11} - \frac{1}{10} a^{10} - \frac{1}{5} a^{9} + \frac{1}{10} a^{7} - \frac{3}{10} a^{6} - \frac{3}{10} a^{5} + \frac{1}{10} a^{4} + \frac{1}{10} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{260} a^{16} - \frac{1}{130} a^{15} + \frac{3}{65} a^{14} + \frac{2}{13} a^{13} + \frac{31}{130} a^{12} + \frac{4}{65} a^{11} - \frac{5}{26} a^{10} + \frac{1}{10} a^{9} + \frac{14}{65} a^{8} + \frac{1}{13} a^{7} - \frac{37}{130} a^{6} - \frac{22}{65} a^{5} - \frac{3}{52} a^{4} + \frac{18}{65} a^{3} - \frac{28}{65} a^{2} + \frac{24}{65} a + \frac{4}{13}$, $\frac{1}{520} a^{17} + \frac{1}{65} a^{15} + \frac{3}{130} a^{14} + \frac{29}{130} a^{13} - \frac{2}{65} a^{12} + \frac{43}{260} a^{11} + \frac{7}{65} a^{10} + \frac{3}{52} a^{9} - \frac{16}{65} a^{8} - \frac{41}{130} a^{7} - \frac{7}{130} a^{6} - \frac{191}{520} a^{5} - \frac{24}{65} a^{4} - \frac{101}{260} a^{3} + \frac{7}{130} a^{2} - \frac{49}{130} a + \frac{4}{13}$, $\frac{1}{1771120} a^{18} + \frac{817}{885560} a^{17} - \frac{453}{442780} a^{16} - \frac{3017}{88556} a^{15} - \frac{16077}{442780} a^{14} + \frac{42429}{442780} a^{13} - \frac{15741}{177112} a^{12} - \frac{7791}{88556} a^{11} + \frac{165809}{885560} a^{10} + \frac{30649}{221390} a^{9} - \frac{21259}{442780} a^{8} + \frac{5522}{110695} a^{7} + \frac{71057}{1771120} a^{6} - \frac{16179}{68120} a^{5} - \frac{407449}{885560} a^{4} - \frac{59227}{442780} a^{3} + \frac{43285}{88556} a^{2} + \frac{14197}{221390} a + \frac{5179}{22139}$, $\frac{1}{7973924968841613354483040} a^{19} + \frac{677717625051966329}{3986962484420806677241520} a^{18} - \frac{2784450570573196124}{19168088867407724409815} a^{17} + \frac{3086892906570412704257}{1993481242210403338620760} a^{16} - \frac{48454580373208481143977}{1993481242210403338620760} a^{15} + \frac{44707004260238592542327}{1993481242210403338620760} a^{14} - \frac{450669142888865145806433}{3986962484420806677241520} a^{13} - \frac{155202725590594489767853}{1993481242210403338620760} a^{12} - \frac{88483961816127475249387}{3986962484420806677241520} a^{11} + \frac{4024192439617816187079}{99674062110520166931038} a^{10} + \frac{99493107400677226181467}{1993481242210403338620760} a^{9} - \frac{126617834927658993061959}{996740621105201669310380} a^{8} - \frac{2818017256456809086078847}{7973924968841613354483040} a^{7} - \frac{1031810477830757681019003}{3986962484420806677241520} a^{6} - \frac{1638399150765970119791631}{3986962484420806677241520} a^{5} - \frac{518473979082371143056903}{1993481242210403338620760} a^{4} + \frac{579979709313573711801987}{1993481242210403338620760} a^{3} + \frac{358804650672799933597543}{996740621105201669310380} a^{2} + \frac{9091853983535667520413}{19168088867407724409815} a + \frac{4398091836987065941807}{49837031055260083465519}$

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:  \( 478264.707493 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-122}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{-2}, \sqrt{61})\), 5.1.238144.1 x5, 10.0.27675731591168.1, 10.0.453700517888.1 x5, 10.2.3459466448896.1 x5

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

Sibling fields

Degree 10 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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$2$2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$61$61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$