↑

Properties

Label 20.6.36709010990...4375.1
Degree $20$
Signature $[6, 7]$
Discriminant $-\,5^{11}\cdot 13^{12}\cdot 19^{9}$
Root discriminant $42.49$
Ramified primes $5, 13, 19$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T633

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8525, -37500, -73115, -81440, -53136, -15256, 5008, 6542, 2364, 601, 908, 961, 459, -17, -114, 15, 42, -7, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 12*x^18 - 7*x^17 + 42*x^16 + 15*x^15 - 114*x^14 - 17*x^13 + 459*x^12 + 961*x^11 + 908*x^10 + 601*x^9 + 2364*x^8 + 6542*x^7 + 5008*x^6 - 15256*x^5 - 53136*x^4 - 81440*x^3 - 73115*x^2 - 37500*x - 8525)
 
gp: K = bnfinit(x^20 - 12*x^18 - 7*x^17 + 42*x^16 + 15*x^15 - 114*x^14 - 17*x^13 + 459*x^12 + 961*x^11 + 908*x^10 + 601*x^9 + 2364*x^8 + 6542*x^7 + 5008*x^6 - 15256*x^5 - 53136*x^4 - 81440*x^3 - 73115*x^2 - 37500*x - 8525, 1)
 

Normalized defining polynomial

\( x^{20} - 12 x^{18} - 7 x^{17} + 42 x^{16} + 15 x^{15} - 114 x^{14} - 17 x^{13} + 459 x^{12} + 961 x^{11} + 908 x^{10} + 601 x^{9} + 2364 x^{8} + 6542 x^{7} + 5008 x^{6} - 15256 x^{5} - 53136 x^{4} - 81440 x^{3} - 73115 x^{2} - 37500 x - 8525 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-367090109903933844075668896484375=-\,5^{11}\cdot 13^{12}\cdot 19^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.49$
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:  $5, 13, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{70} a^{18} - \frac{1}{10} a^{16} - \frac{1}{10} a^{15} - \frac{2}{5} a^{14} + \frac{3}{14} a^{13} + \frac{13}{35} a^{12} + \frac{23}{70} a^{11} - \frac{3}{35} a^{10} + \frac{13}{35} a^{9} - \frac{16}{35} a^{8} - \frac{2}{35} a^{7} - \frac{1}{70} a^{6} - \frac{23}{70} a^{5} - \frac{1}{35} a^{4} - \frac{3}{35} a^{3} - \frac{8}{35} a^{2} - \frac{5}{14} a - \frac{1}{7}$, $\frac{1}{25997614609340880775693308710} a^{19} + \frac{27603479811092668306033137}{12998807304670440387846654355} a^{18} - \frac{323713830915486267322058061}{3713944944191554396527615530} a^{17} + \frac{135892949080014121442970105}{742788988838310879305523106} a^{16} + \frac{443406862197034631497610541}{1856972472095777198263807765} a^{15} - \frac{640598101242861912654411067}{25997614609340880775693308710} a^{14} + \frac{726618288200878925838614148}{12998807304670440387846654355} a^{13} - \frac{1279792923012892098703186379}{3713944944191554396527615530} a^{12} - \frac{4037775255806148192928618567}{12998807304670440387846654355} a^{11} - \frac{2754822798689877335742654294}{12998807304670440387846654355} a^{10} + \frac{1430297398921803586870922881}{12998807304670440387846654355} a^{9} - \frac{5384438311685453931490593206}{12998807304670440387846654355} a^{8} + \frac{8246641994171344860512730393}{25997614609340880775693308710} a^{7} - \frac{11306289877974542228620249137}{25997614609340880775693308710} a^{6} - \frac{191913108280578866693613922}{12998807304670440387846654355} a^{5} - \frac{4939869038869543617435676062}{12998807304670440387846654355} a^{4} - \frac{1037180703426624565511426848}{2599761460934088077569330871} a^{3} - \frac{9977071532212300593152231379}{25997614609340880775693308710} a^{2} - \frac{148779118563158977663226517}{371394494419155439652761553} a + \frac{365779171629428004285424544}{2599761460934088077569330871}$

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:  $12$
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:  \( 227609658.024 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T633:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40960
The 124 conjugacy class representatives for t20n633 are not computed
Character table for t20n633 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.19827925.1, 10.10.1965733049028125.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R $20$ R $20$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
13Data not computed
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$