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Properties

Label 20.12.1709518871...4121.1
Degree $20$
Signature $[12, 4]$
Discriminant $13^{2}\cdot 97^{2}\cdot 401^{10}$
Root discriminant $40.89$
Ramified primes $13, 97, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T347

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![729, 3807, -1980, -29406, -9937, 68268, 33555, -60030, -29882, 13205, 10834, 9101, -3611, -4351, 938, 482, -43, 2, -14, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 14*x^18 + 2*x^17 - 43*x^16 + 482*x^15 + 938*x^14 - 4351*x^13 - 3611*x^12 + 9101*x^11 + 10834*x^10 + 13205*x^9 - 29882*x^8 - 60030*x^7 + 33555*x^6 + 68268*x^5 - 9937*x^4 - 29406*x^3 - 1980*x^2 + 3807*x + 729)
 
gp: K = bnfinit(x^20 - x^19 - 14*x^18 + 2*x^17 - 43*x^16 + 482*x^15 + 938*x^14 - 4351*x^13 - 3611*x^12 + 9101*x^11 + 10834*x^10 + 13205*x^9 - 29882*x^8 - 60030*x^7 + 33555*x^6 + 68268*x^5 - 9937*x^4 - 29406*x^3 - 1980*x^2 + 3807*x + 729, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 14 x^{18} + 2 x^{17} - 43 x^{16} + 482 x^{15} + 938 x^{14} - 4351 x^{13} - 3611 x^{12} + 9101 x^{11} + 10834 x^{10} + 13205 x^{9} - 29882 x^{8} - 60030 x^{7} + 33555 x^{6} + 68268 x^{5} - 9937 x^{4} - 29406 x^{3} - 1980 x^{2} + 3807 x + 729 \)

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:  \(170951887142655083512160513274121=13^{2}\cdot 97^{2}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.89$
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:  $13, 97, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{27} a^{16} - \frac{4}{27} a^{15} - \frac{1}{9} a^{14} - \frac{7}{27} a^{12} + \frac{8}{27} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{10}{27} a^{8} + \frac{5}{27} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{5}{27} a^{4} - \frac{1}{9} a^{3} + \frac{8}{27} a^{2} + \frac{4}{9} a$, $\frac{1}{27} a^{17} - \frac{1}{27} a^{15} - \frac{1}{9} a^{14} + \frac{2}{27} a^{13} - \frac{2}{27} a^{12} + \frac{8}{27} a^{11} + \frac{2}{9} a^{10} - \frac{5}{27} a^{9} + \frac{2}{27} a^{7} + \frac{13}{27} a^{5} + \frac{4}{27} a^{4} + \frac{5}{27} a^{3} - \frac{10}{27} a^{2} - \frac{2}{9} a$, $\frac{1}{2241} a^{18} + \frac{40}{2241} a^{17} - \frac{20}{2241} a^{16} + \frac{80}{747} a^{15} - \frac{277}{2241} a^{14} + \frac{68}{747} a^{13} - \frac{884}{2241} a^{12} + \frac{64}{747} a^{11} - \frac{479}{2241} a^{10} - \frac{131}{2241} a^{9} + \frac{334}{2241} a^{8} - \frac{194}{747} a^{7} - \frac{1094}{2241} a^{6} + \frac{1010}{2241} a^{5} - \frac{28}{2241} a^{4} + \frac{418}{2241} a^{3} - \frac{55}{249} a^{2} + \frac{71}{249} a + \frac{4}{83}$, $\frac{1}{8146206581014469692922420694316847409} a^{19} + \frac{490406142878679887188038556338164}{8146206581014469692922420694316847409} a^{18} + \frac{38256742211935885810140333463321006}{8146206581014469692922420694316847409} a^{17} - \frac{146499399871353350550352173248742067}{8146206581014469692922420694316847409} a^{16} - \frac{235957864189877233130251729792685089}{8146206581014469692922420694316847409} a^{15} - \frac{778699690958611126992453574599719614}{8146206581014469692922420694316847409} a^{14} + \frac{1211522503697370912486423906962565284}{8146206581014469692922420694316847409} a^{13} + \frac{545376764964630535579776356242527068}{8146206581014469692922420694316847409} a^{12} + \frac{2546123508420050631156209859412791940}{8146206581014469692922420694316847409} a^{11} + \frac{1799400780144965535626114811630479339}{8146206581014469692922420694316847409} a^{10} - \frac{3518738862736911089248728080104523873}{8146206581014469692922420694316847409} a^{9} + \frac{2705828044171098015920672843064742802}{8146206581014469692922420694316847409} a^{8} + \frac{286138561842444402860998778599195642}{8146206581014469692922420694316847409} a^{7} + \frac{444173205116804104204088224482476171}{2715402193671489897640806898105615803} a^{6} - \frac{1144553440484354829972327454859798917}{2715402193671489897640806898105615803} a^{5} + \frac{123051289534831613060833657529684905}{301711354852387766404534099789512867} a^{4} + \frac{3517333519550980342724914466913125936}{8146206581014469692922420694316847409} a^{3} + \frac{762568697010055135619532444155437618}{2715402193671489897640806898105615803} a^{2} + \frac{42998330225309873857446716708330868}{100570451617462588801511366596504289} a - \frac{8751240779502688427454907128449033}{100570451617462588801511366596504289}$

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:  \( 1179350412.82 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T347:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 104 conjugacy class representatives for t20n347 are not computed
Character table for t20n347 is not computed

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.10.10368641602001.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
401Data not computed