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Properties

Label 20.12.1606056182...5712.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{34}\cdot 2657^{7}$
Root discriminant $51.32$
Ramified primes $2, 2657$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T547

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-64, 2816, -12576, -32512, 40636, 75968, -27384, -39492, -18688, -41708, 637, 30916, 13040, -6682, -4513, 472, 647, -2, -44, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 44*x^18 - 2*x^17 + 647*x^16 + 472*x^15 - 4513*x^14 - 6682*x^13 + 13040*x^12 + 30916*x^11 + 637*x^10 - 41708*x^9 - 18688*x^8 - 39492*x^7 - 27384*x^6 + 75968*x^5 + 40636*x^4 - 32512*x^3 - 12576*x^2 + 2816*x - 64)
 
gp: K = bnfinit(x^20 - 44*x^18 - 2*x^17 + 647*x^16 + 472*x^15 - 4513*x^14 - 6682*x^13 + 13040*x^12 + 30916*x^11 + 637*x^10 - 41708*x^9 - 18688*x^8 - 39492*x^7 - 27384*x^6 + 75968*x^5 + 40636*x^4 - 32512*x^3 - 12576*x^2 + 2816*x - 64, 1)
 

Normalized defining polynomial

\( x^{20} - 44 x^{18} - 2 x^{17} + 647 x^{16} + 472 x^{15} - 4513 x^{14} - 6682 x^{13} + 13040 x^{12} + 30916 x^{11} + 637 x^{10} - 41708 x^{9} - 18688 x^{8} - 39492 x^{7} - 27384 x^{6} + 75968 x^{5} + 40636 x^{4} - 32512 x^{3} - 12576 x^{2} + 2816 x - 64 \)

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:  \(16060561821304241463001418628595712=2^{34}\cdot 2657^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.32$
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:  $2, 2657$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{11} + \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{13} + \frac{1}{8} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} - \frac{3}{8} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{704} a^{18} - \frac{1}{16} a^{17} + \frac{3}{88} a^{16} - \frac{1}{352} a^{15} + \frac{79}{704} a^{14} + \frac{7}{176} a^{13} - \frac{21}{704} a^{12} - \frac{79}{352} a^{11} + \frac{21}{176} a^{10} + \frac{27}{176} a^{9} + \frac{13}{704} a^{8} - \frac{1}{8} a^{7} - \frac{51}{176} a^{6} - \frac{17}{176} a^{5} + \frac{13}{88} a^{4} - \frac{7}{44} a^{3} + \frac{47}{176} a^{2} + \frac{1}{4} a + \frac{13}{44}$, $\frac{1}{58345022722925180451247059026776533158528} a^{19} + \frac{9528571868845678327219930488095449301}{29172511361462590225623529513388266579264} a^{18} - \frac{76091096429487166444226230516095159337}{1823281960091411889101470594586766661204} a^{17} + \frac{1846024459777923733850443227724627330335}{29172511361462590225623529513388266579264} a^{16} - \frac{5127251701640516243225625336070884160333}{58345022722925180451247059026776533158528} a^{15} - \frac{2581070906698178035750316141539991159421}{29172511361462590225623529513388266579264} a^{14} - \frac{5579563264635058881856362131303860534125}{58345022722925180451247059026776533158528} a^{13} + \frac{64584949367933167770862672068890342387}{1326023243702845010255614977881284844512} a^{12} + \frac{480655451678962676051922090961811832709}{3646563920182823778202941189173533322408} a^{11} + \frac{2753346805967867940206271375711747113797}{14586255680731295112811764756694133289632} a^{10} + \frac{1949412454889597160358565780266851456613}{58345022722925180451247059026776533158528} a^{9} + \frac{9511144595055836983748942666271457907635}{29172511361462590225623529513388266579264} a^{8} - \frac{1124784115307056067124388506614604878087}{14586255680731295112811764756694133289632} a^{7} + \frac{3830377660037742048385510754859481427905}{14586255680731295112811764756694133289632} a^{6} + \frac{81537686134278563351038464831682381597}{3646563920182823778202941189173533322408} a^{5} - \frac{229226125288781699883422616548331858679}{911640980045705944550735297293383330602} a^{4} + \frac{4882259288262689714876978506118311208015}{14586255680731295112811764756694133289632} a^{3} + \frac{678653466054371099102651342534418879323}{7293127840365647556405882378347066644816} a^{2} - \frac{950304678488306783726293757305614659845}{3646563920182823778202941189173533322408} a - \frac{266716688582304817103138632745513056577}{1823281960091411889101470594586766661204}$

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:  \( 44690399614.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T547:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 28800
The 41 conjugacy class representatives for t20n547
Character table for t20n547 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.170048.1, 10.6.925322313728.1

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

Sibling fields

Degree 20 sibling: data not computed
Degree 24 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 $20$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$2$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.6.11.1$x^{6} + 14$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
2.6.11.1$x^{6} + 14$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
2657Data not computed