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Properties

Label 20.12.6400925851...0112.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{44}\cdot 439\cdot 1663\cdot 2657^{4}$
Root discriminant $43.68$
Ramified primes $2, 439, 1663, 2657$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1036

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-178, -632, 1544, 7400, 888, -21780, -14684, 27136, 26242, -17120, -21342, 5840, 9519, -1088, -2472, 104, 372, -4, -30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 30*x^18 - 4*x^17 + 372*x^16 + 104*x^15 - 2472*x^14 - 1088*x^13 + 9519*x^12 + 5840*x^11 - 21342*x^10 - 17120*x^9 + 26242*x^8 + 27136*x^7 - 14684*x^6 - 21780*x^5 + 888*x^4 + 7400*x^3 + 1544*x^2 - 632*x - 178)
 
gp: K = bnfinit(x^20 - 30*x^18 - 4*x^17 + 372*x^16 + 104*x^15 - 2472*x^14 - 1088*x^13 + 9519*x^12 + 5840*x^11 - 21342*x^10 - 17120*x^9 + 26242*x^8 + 27136*x^7 - 14684*x^6 - 21780*x^5 + 888*x^4 + 7400*x^3 + 1544*x^2 - 632*x - 178, 1)
 

Normalized defining polynomial

\( x^{20} - 30 x^{18} - 4 x^{17} + 372 x^{16} + 104 x^{15} - 2472 x^{14} - 1088 x^{13} + 9519 x^{12} + 5840 x^{11} - 21342 x^{10} - 17120 x^{9} + 26242 x^{8} + 27136 x^{7} - 14684 x^{6} - 21780 x^{5} + 888 x^{4} + 7400 x^{3} + 1544 x^{2} - 632 x - 178 \)

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:  \(640092585108940580184055833690112=2^{44}\cdot 439\cdot 1663\cdot 2657^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.68$
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:  $2, 439, 1663, 2657$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{12}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{13}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{14}$, $\frac{1}{91225716643702867210} a^{19} + \frac{7652991240354163531}{45612858321851433605} a^{18} - \frac{15352872638846815741}{91225716643702867210} a^{17} + \frac{15820283820311773179}{91225716643702867210} a^{16} + \frac{7824681734768921227}{18245143328740573442} a^{15} + \frac{18850450760774589622}{45612858321851433605} a^{14} - \frac{14087065620576525289}{91225716643702867210} a^{13} - \frac{22939902919077096501}{91225716643702867210} a^{12} + \frac{10905146968938952626}{45612858321851433605} a^{11} + \frac{16282540316413025047}{45612858321851433605} a^{10} + \frac{4081601448374959488}{45612858321851433605} a^{9} + \frac{18500200206988455386}{45612858321851433605} a^{8} + \frac{5443729043903212663}{45612858321851433605} a^{7} + \frac{6815489575582177494}{45612858321851433605} a^{6} + \frac{231654383609066041}{45612858321851433605} a^{5} + \frac{18013973445743802482}{45612858321851433605} a^{4} - \frac{7067147884542023237}{45612858321851433605} a^{3} + \frac{21243728978906168956}{45612858321851433605} a^{2} + \frac{7889652323279880494}{45612858321851433605} a - \frac{13170160679792550213}{45612858321851433605}$

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:  \( 3633107511.73 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1036:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 396 conjugacy class representatives for t20n1036 are not computed
Character table for t20n1036 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 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.16.13$x^{8} + 6 x^{6} + 4 x^{5} + 2 x^{4} + 4$$4$$2$$16$$D_4\times C_2$$[2, 2, 3]^{2}$
2.12.28.225$x^{12} + 4 x^{11} + 2 x^{10} + 2 x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{2} - 2$$12$$1$$28$$C_2 \times S_4$$[8/3, 8/3, 3]_{3}^{2}$
439Data not computed
1663Data not computed
2657Data not computed