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Properties

Label 20.8.34136437243...2944.2
Degree $20$
Signature $[8, 6]$
Discriminant $2^{40}\cdot 11^{18}\cdot 89^{5}$
Root discriminant $106.33$
Ramified primes $2, 11, 89$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group 20T310

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19911881, 0, -79682614, 0, -11841665, 0, 16052102, 0, 2648932, 0, -702064, 0, -140151, 0, 396, 0, 1199, 0, 66, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 66*x^18 + 1199*x^16 + 396*x^14 - 140151*x^12 - 702064*x^10 + 2648932*x^8 + 16052102*x^6 - 11841665*x^4 - 79682614*x^2 + 19911881)
 
gp: K = bnfinit(x^20 + 66*x^18 + 1199*x^16 + 396*x^14 - 140151*x^12 - 702064*x^10 + 2648932*x^8 + 16052102*x^6 - 11841665*x^4 - 79682614*x^2 + 19911881, 1)
 

Normalized defining polynomial

\( x^{20} + 66 x^{18} + 1199 x^{16} + 396 x^{14} - 140151 x^{12} - 702064 x^{10} + 2648932 x^{8} + 16052102 x^{6} - 11841665 x^{4} - 79682614 x^{2} + 19911881 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(34136437243167592795589550820145909202944=2^{40}\cdot 11^{18}\cdot 89^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $106.33$
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:  $2, 11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11} a^{10}$, $\frac{1}{11} a^{11}$, $\frac{1}{11} a^{12}$, $\frac{1}{11} a^{13}$, $\frac{1}{11} a^{14}$, $\frac{1}{11} a^{15}$, $\frac{1}{121} a^{16} + \frac{4}{11} a^{8} - \frac{1}{11} a^{6}$, $\frac{1}{121} a^{17} + \frac{4}{11} a^{9} - \frac{1}{11} a^{7}$, $\frac{1}{329123807066965037417487029322083} a^{18} + \frac{15678150168160280544260064249}{329123807066965037417487029322083} a^{16} + \frac{8343003657906348376853472348}{29920346096996821583407911756553} a^{14} + \frac{566565172183953054592434291310}{29920346096996821583407911756553} a^{12} - \frac{99794710249363090037016729136}{29920346096996821583407911756553} a^{10} + \frac{751236886977485241031152180619}{29920346096996821583407911756553} a^{8} - \frac{11357372680990315630017620978899}{29920346096996821583407911756553} a^{6} - \frac{38492113710022855036807096497}{118262237537536844203193327101} a^{4} + \frac{3712800444739868724889833408}{118262237537536844203193327101} a^{2} - \frac{129643278016156841911096296225}{2720031463363347416673446523323}$, $\frac{1}{14152323703879496608951942260849569} a^{19} + \frac{15678150168160280544260064249}{14152323703879496608951942260849569} a^{17} + \frac{35368752027381422765131658275547}{1286574882170863328086540205531779} a^{15} - \frac{21193686534722826278795137895274}{1286574882170863328086540205531779} a^{13} + \frac{18940425533294068826677108934125}{1286574882170863328086540205531779} a^{11} - \frac{298452224082990730593047965384911}{1286574882170863328086540205531779} a^{9} - \frac{549923602426933104131360032596853}{1286574882170863328086540205531779} a^{7} + \frac{1262392499202882431198319501614}{5085276214114084300737313065343} a^{5} - \frac{2006745237693386482729396727309}{5085276214114084300737313065343} a^{3} - \frac{40930115228466368092012794146070}{116961352924623938916958200502889} a$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  $13$
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:  \( 859423529927 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T310:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 44 conjugacy class representatives for t20n310
Character table for t20n310 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.1738687177114624.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
89Data not computed