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Properties

Label 20.12.2549631546...0944.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{54}\cdot 73^{4}\cdot 2657^{4}$
Root discriminant $74.19$
Ramified primes $2, 73, 2657$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1013

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![113592964, 0, -199176704, 0, 122407130, 0, -33525396, 0, 3670193, 0, 52254, 0, -28999, 0, -2364, 0, 699, 0, -46, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 46*x^18 + 699*x^16 - 2364*x^14 - 28999*x^12 + 52254*x^10 + 3670193*x^8 - 33525396*x^6 + 122407130*x^4 - 199176704*x^2 + 113592964)
 
gp: K = bnfinit(x^20 - 46*x^18 + 699*x^16 - 2364*x^14 - 28999*x^12 + 52254*x^10 + 3670193*x^8 - 33525396*x^6 + 122407130*x^4 - 199176704*x^2 + 113592964, 1)
 

Normalized defining polynomial

\( x^{20} - 46 x^{18} + 699 x^{16} - 2364 x^{14} - 28999 x^{12} + 52254 x^{10} + 3670193 x^{8} - 33525396 x^{6} + 122407130 x^{4} - 199176704 x^{2} + 113592964 \)

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:  \(25496315463173953254560344368631250944=2^{54}\cdot 73^{4}\cdot 2657^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.19$
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:  $2, 73, 2657$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{146} a^{14} + \frac{27}{146} a^{12} - \frac{31}{146} a^{10} + \frac{45}{146} a^{8} - \frac{9}{73} a^{6} - \frac{7}{73} a^{4} - \frac{14}{73} a^{2}$, $\frac{1}{146} a^{15} + \frac{27}{146} a^{13} - \frac{31}{146} a^{11} + \frac{45}{146} a^{9} - \frac{9}{73} a^{7} - \frac{7}{73} a^{5} - \frac{14}{73} a^{3}$, $\frac{1}{10658} a^{16} + \frac{27}{10658} a^{14} - \frac{2659}{10658} a^{12} + \frac{351}{5329} a^{10} + \frac{931}{10658} a^{8} + \frac{2979}{10658} a^{6} + \frac{2819}{10658} a^{4} + \frac{18}{73} a^{2}$, $\frac{1}{10658} a^{17} + \frac{27}{10658} a^{15} - \frac{2659}{10658} a^{13} + \frac{351}{5329} a^{11} + \frac{931}{10658} a^{9} + \frac{2979}{10658} a^{7} + \frac{2819}{10658} a^{5} + \frac{18}{73} a^{3}$, $\frac{1}{1719822754195971493825348598} a^{18} + \frac{42323120243539510539077}{1719822754195971493825348598} a^{16} - \frac{2290620081424650222001574}{859911377097985746912674299} a^{14} - \frac{195975016810610820310983145}{859911377097985746912674299} a^{12} + \frac{164952495637166167726086051}{1719822754195971493825348598} a^{10} + \frac{630942941037678219015047513}{1719822754195971493825348598} a^{8} + \frac{334653506841998686369703374}{859911377097985746912674299} a^{6} + \frac{3838172583259091067458676}{11779607905451859546748963} a^{4} - \frac{50284334031407569009070}{161364491855504925297931} a^{2} - \frac{130496792331818843491}{2210472491171300346547}$, $\frac{1}{3439645508391942987650697196} a^{19} - \frac{1}{3439645508391942987650697196} a^{18} + \frac{42323120243539510539077}{3439645508391942987650697196} a^{17} - \frac{42323120243539510539077}{3439645508391942987650697196} a^{16} - \frac{1145310040712325111000787}{859911377097985746912674299} a^{15} + \frac{1145310040712325111000787}{859911377097985746912674299} a^{14} - \frac{195975016810610820310983145}{1719822754195971493825348598} a^{13} + \frac{195975016810610820310983145}{1719822754195971493825348598} a^{12} + \frac{164952495637166167726086051}{3439645508391942987650697196} a^{11} - \frac{164952495637166167726086051}{3439645508391942987650697196} a^{10} + \frac{630942941037678219015047513}{3439645508391942987650697196} a^{9} - \frac{630942941037678219015047513}{3439645508391942987650697196} a^{8} + \frac{167326753420999343184851687}{859911377097985746912674299} a^{7} - \frac{167326753420999343184851687}{859911377097985746912674299} a^{6} + \frac{1919086291629545533729338}{11779607905451859546748963} a^{5} - \frac{1919086291629545533729338}{11779607905451859546748963} a^{4} - \frac{25142167015703784504535}{161364491855504925297931} a^{3} + \frac{25142167015703784504535}{161364491855504925297931} a^{2} + \frac{1039987849419740751528}{2210472491171300346547} a - \frac{1039987849419740751528}{2210472491171300346547}$

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:  \( 609161460134 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1013:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3686400
The 114 conjugacy class representatives for t20n1013 are not computed
Character table for t20n1013 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 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 32 sibling: 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$2.8.22.62$x^{8} + 8 x^{5} + 10 x^{4} + 4$$4$$2$$22$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 2, 3, 7/2, 4]^{2}$
2.12.32.289$x^{12} + 4 x^{11} - 4 x^{10} - 4 x^{9} + 4 x^{6} + 8 x^{5} + 6 x^{4} + 8 x^{3} - 4 x^{2} + 8 x - 2$$12$$1$$32$12T193$[4/3, 4/3, 2, 3, 19/6, 19/6, 7/2]_{3}^{2}$
73Data not computed
2657Data not computed