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Properties

Label 20.16.1214556712...8125.1
Degree $20$
Signature $[16, 2]$
Discriminant $3^{4}\cdot 5^{15}\cdot 23^{8}\cdot 89^{4}$
Root discriminant $35.83$
Ramified primes $3, 5, 23, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T802

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41, -311, 874, 4409, -9757, -14678, 30545, 17434, -37564, -16632, 32434, 5527, -16124, 828, 4112, -827, -476, 160, 13, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 13*x^18 + 160*x^17 - 476*x^16 - 827*x^15 + 4112*x^14 + 828*x^13 - 16124*x^12 + 5527*x^11 + 32434*x^10 - 16632*x^9 - 37564*x^8 + 17434*x^7 + 30545*x^6 - 14678*x^5 - 9757*x^4 + 4409*x^3 + 874*x^2 - 311*x + 41)
 
gp: K = bnfinit(x^20 - 10*x^19 + 13*x^18 + 160*x^17 - 476*x^16 - 827*x^15 + 4112*x^14 + 828*x^13 - 16124*x^12 + 5527*x^11 + 32434*x^10 - 16632*x^9 - 37564*x^8 + 17434*x^7 + 30545*x^6 - 14678*x^5 - 9757*x^4 + 4409*x^3 + 874*x^2 - 311*x + 41, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 13 x^{18} + 160 x^{17} - 476 x^{16} - 827 x^{15} + 4112 x^{14} + 828 x^{13} - 16124 x^{12} + 5527 x^{11} + 32434 x^{10} - 16632 x^{9} - 37564 x^{8} + 17434 x^{7} + 30545 x^{6} - 14678 x^{5} - 9757 x^{4} + 4409 x^{3} + 874 x^{2} - 311 x + 41 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12145567127297495495635986328125=3^{4}\cdot 5^{15}\cdot 23^{8}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.83$
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:  $3, 5, 23, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{3} a^{18} + \frac{1}{3} a^{15} - \frac{1}{3} a^{14} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{4883742663332458871142762717} a^{19} + \frac{509586836404760851742117527}{4883742663332458871142762717} a^{18} + \frac{675661221714517865276668972}{1627914221110819623714254239} a^{17} - \frac{914906912605121881365473546}{4883742663332458871142762717} a^{16} + \frac{17218427216299311836893702}{1627914221110819623714254239} a^{15} + \frac{2318349255839610752345943784}{4883742663332458871142762717} a^{14} + \frac{2314775642205637457492371057}{4883742663332458871142762717} a^{13} - \frac{1512146896237362006349328998}{4883742663332458871142762717} a^{12} - \frac{677598877600850544028673134}{4883742663332458871142762717} a^{11} + \frac{1373014445687271852472261358}{4883742663332458871142762717} a^{10} - \frac{2407732316025963131450723050}{4883742663332458871142762717} a^{9} - \frac{1088183860438591215281992304}{4883742663332458871142762717} a^{8} - \frac{264635695921015841002484522}{4883742663332458871142762717} a^{7} - \frac{35883815340363914064294184}{1627914221110819623714254239} a^{6} + \frac{436916833299325482391855637}{4883742663332458871142762717} a^{5} + \frac{221448114114009019069847}{4883742663332458871142762717} a^{4} - \frac{254653249161180864643322444}{1627914221110819623714254239} a^{3} - \frac{412077649751861091693911152}{4883742663332458871142762717} a^{2} + \frac{569526838300206770978416160}{4883742663332458871142762717} a - \frac{47753897216749636945137065}{119115674715425826125433237}$

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:  $17$
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:  \( 622054132.862 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T802:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 138 conjugacy class representatives for t20n802 are not computed
Character table for t20n802 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.767625.1, 10.10.2946240703125.1

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 $20$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$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}$
3.12.0.1$x^{12} - x^{4} - x^{3} - x^{2} + x - 1$$1$$12$$0$$C_{12}$$[\ ]^{12}$
5Data not computed
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.8.6.2$x^{8} - 1633 x^{4} + 1270129$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$89$89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.6.0.1$x^{6} - x + 6$$1$$6$$0$$C_6$$[\ ]^{6}$
89.6.0.1$x^{6} - x + 6$$1$$6$$0$$C_6$$[\ ]^{6}$