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Properties

Label 20.12.7063888397...9408.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{30}\cdot 3^{15}\cdot 71^{9}$
Root discriminant $43.90$
Ramified primes $2, 3, 71$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T647

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-359, 4208, -18846, 37256, -10204, -106766, 233064, -200536, 23639, 87666, -66624, 12156, 11518, -8476, 1512, 742, -456, 64, 18, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 18*x^18 + 64*x^17 - 456*x^16 + 742*x^15 + 1512*x^14 - 8476*x^13 + 11518*x^12 + 12156*x^11 - 66624*x^10 + 87666*x^9 + 23639*x^8 - 200536*x^7 + 233064*x^6 - 106766*x^5 - 10204*x^4 + 37256*x^3 - 18846*x^2 + 4208*x - 359)
 
gp: K = bnfinit(x^20 - 8*x^19 + 18*x^18 + 64*x^17 - 456*x^16 + 742*x^15 + 1512*x^14 - 8476*x^13 + 11518*x^12 + 12156*x^11 - 66624*x^10 + 87666*x^9 + 23639*x^8 - 200536*x^7 + 233064*x^6 - 106766*x^5 - 10204*x^4 + 37256*x^3 - 18846*x^2 + 4208*x - 359, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 18 x^{18} + 64 x^{17} - 456 x^{16} + 742 x^{15} + 1512 x^{14} - 8476 x^{13} + 11518 x^{12} + 12156 x^{11} - 66624 x^{10} + 87666 x^{9} + 23639 x^{8} - 200536 x^{7} + 233064 x^{6} - 106766 x^{5} - 10204 x^{4} + 37256 x^{3} - 18846 x^{2} + 4208 x - 359 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(706388839731582814105670315409408=2^{30}\cdot 3^{15}\cdot 71^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.90$
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:  $2, 3, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{1802740975937164011697187066592593876281} a^{19} - \frac{608316183495526936346138349522775583095}{1802740975937164011697187066592593876281} a^{18} + \frac{647378527754756779955755992467431561791}{1802740975937164011697187066592593876281} a^{17} + \frac{901152783270892426698816620869728264448}{1802740975937164011697187066592593876281} a^{16} + \frac{641442137212339322599056646027527962602}{1802740975937164011697187066592593876281} a^{15} + \frac{32443780838155591655442376094434744425}{138672382764397231669014389737891836637} a^{14} + \frac{65734703238282199090296410981492294600}{1802740975937164011697187066592593876281} a^{13} + \frac{5816493020316369837055345110966503322}{16538908036120770749515477675161411709} a^{12} - \frac{429033612280390706344214645603813038353}{1802740975937164011697187066592593876281} a^{11} + \frac{163503014433002448952846900339823243774}{1802740975937164011697187066592593876281} a^{10} - \frac{756582005523193325215189865634378463872}{1802740975937164011697187066592593876281} a^{9} - \frac{508952873998455685450435604098832537903}{1802740975937164011697187066592593876281} a^{8} + \frac{6739327451647584731518935751751706940}{16538908036120770749515477675161411709} a^{7} + \frac{31256170415055735771280154597970374638}{138672382764397231669014389737891836637} a^{6} + \frac{30491677053093673175820100700085396676}{138672382764397231669014389737891836637} a^{5} + \frac{419484716505886168197268777551600304976}{1802740975937164011697187066592593876281} a^{4} + \frac{414810093267414050152135591802122900786}{1802740975937164011697187066592593876281} a^{3} + \frac{37625246645672506151902718196723526186}{138672382764397231669014389737891836637} a^{2} - \frac{184231811908037525410544205389154736670}{1802740975937164011697187066592593876281} a + \frac{431836472604419032965744762618329773585}{1802740975937164011697187066592593876281}$

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:  $15$
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:  \( 1594259308.02 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T647:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 51200
The 152 conjugacy class representatives for t20n647 are not computed
Character table for t20n647 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 10.10.6323239406592.1

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 $20$ $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
$71$71.5.0.1$x^{5} - x + 8$$1$$5$$0$$C_5$$[\ ]^{5}$
71.5.0.1$x^{5} - x + 8$$1$$5$$0$$C_5$$[\ ]^{5}$
71.10.9.5$x^{10} - 18176$$10$$1$$9$$C_{10}$$[\ ]_{10}$