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Properties

Label 20.20.2816215459...3249.1
Degree $20$
Signature $[20, 0]$
Discriminant $3^{10}\cdot 61^{10}\cdot 401^{8}$
Root discriminant $148.76$
Ramified primes $3, 61, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_5\times A_4$ (as 20T37)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![873, 5958, -9972, -67485, 54133, 260630, -160387, -406080, 203500, 305362, -129120, -121756, 44550, 26624, -8616, -3140, 916, 183, -49, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 49*x^18 + 183*x^17 + 916*x^16 - 3140*x^15 - 8616*x^14 + 26624*x^13 + 44550*x^12 - 121756*x^11 - 129120*x^10 + 305362*x^9 + 203500*x^8 - 406080*x^7 - 160387*x^6 + 260630*x^5 + 54133*x^4 - 67485*x^3 - 9972*x^2 + 5958*x + 873)
 
gp: K = bnfinit(x^20 - 4*x^19 - 49*x^18 + 183*x^17 + 916*x^16 - 3140*x^15 - 8616*x^14 + 26624*x^13 + 44550*x^12 - 121756*x^11 - 129120*x^10 + 305362*x^9 + 203500*x^8 - 406080*x^7 - 160387*x^6 + 260630*x^5 + 54133*x^4 - 67485*x^3 - 9972*x^2 + 5958*x + 873, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 49 x^{18} + 183 x^{17} + 916 x^{16} - 3140 x^{15} - 8616 x^{14} + 26624 x^{13} + 44550 x^{12} - 121756 x^{11} - 129120 x^{10} + 305362 x^{9} + 203500 x^{8} - 406080 x^{7} - 160387 x^{6} + 260630 x^{5} + 54133 x^{4} - 67485 x^{3} - 9972 x^{2} + 5958 x + 873 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(28162154599151343753050158965307922722863249=3^{10}\cdot 61^{10}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $148.76$
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:  $3, 61, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{15} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{18} + \frac{1}{3} a^{10} + \frac{1}{3} a^{2}$, $\frac{1}{41374636908488009860424070533772746301} a^{19} + \frac{2120931165694019172727460614837948889}{13791545636162669953474690177924248767} a^{18} + \frac{1360689782352078183466894484322141199}{13791545636162669953474690177924248767} a^{17} + \frac{6690794764396399600072921086807161561}{41374636908488009860424070533772746301} a^{16} + \frac{4507770576068179261001957420060310716}{41374636908488009860424070533772746301} a^{15} - \frac{3378794244451084566938978936043438526}{13791545636162669953474690177924248767} a^{14} + \frac{14919605631736166659597830413777583901}{41374636908488009860424070533772746301} a^{13} + \frac{20380264456871129801259042505628642704}{41374636908488009860424070533772746301} a^{12} - \frac{1821395752309507256612610394737784478}{41374636908488009860424070533772746301} a^{11} + \frac{7465217301623228456869270611790113791}{41374636908488009860424070533772746301} a^{10} + \frac{12425266056678140268279756820058700944}{41374636908488009860424070533772746301} a^{9} - \frac{17461380580499408386702634191354435300}{41374636908488009860424070533772746301} a^{8} + \frac{5777192738957649101435507319575482751}{13791545636162669953474690177924248767} a^{7} - \frac{7109176973569365218038252884059563115}{41374636908488009860424070533772746301} a^{6} - \frac{1731749359138360356830972992604186492}{41374636908488009860424070533772746301} a^{5} - \frac{2169678338325383682166544329662412849}{13791545636162669953474690177924248767} a^{4} + \frac{1265603654781445438628623009917752153}{13791545636162669953474690177924248767} a^{3} + \frac{12719425269287041470930750317713801106}{41374636908488009860424070533772746301} a^{2} + \frac{2676709475501162556377398057949056221}{13791545636162669953474690177924248767} a + \frac{3810615414973359627201241870321447897}{13791545636162669953474690177924248767}$

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:  $19$
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:  \( 3231166245160000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5\times A_4$ (as 20T37):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 16 conjugacy class representatives for $D_5\times A_4$
Character table for $D_5\times A_4$

Intermediate fields

4.4.33489.1, 5.5.160801.1

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

Sibling fields

Degree 30 siblings: data not computed
Degree 40 sibling: 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 $15{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }$ R $15{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ $15{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ $15{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $15{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ $15{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61Data not computed
401Data not computed