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Properties

Label 20.0.15932170210...6016.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{10}\cdot 3^{2}\cdot 401^{12}$
Root discriminant $57.56$
Ramified primes $2, 3, 401$
Class number $2712$ (GRH)
Class group $[2, 2, 678]$ (GRH)
Galois group 20T347

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1025807, -1607848, 1346246, -892260, 1813853, -713742, 774271, -501531, 268289, -182224, 81481, -42411, 16099, -6390, 2749, -516, 363, -19, 29, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 29*x^18 - 19*x^17 + 363*x^16 - 516*x^15 + 2749*x^14 - 6390*x^13 + 16099*x^12 - 42411*x^11 + 81481*x^10 - 182224*x^9 + 268289*x^8 - 501531*x^7 + 774271*x^6 - 713742*x^5 + 1813853*x^4 - 892260*x^3 + 1346246*x^2 - 1607848*x + 1025807)
 
gp: K = bnfinit(x^20 + 29*x^18 - 19*x^17 + 363*x^16 - 516*x^15 + 2749*x^14 - 6390*x^13 + 16099*x^12 - 42411*x^11 + 81481*x^10 - 182224*x^9 + 268289*x^8 - 501531*x^7 + 774271*x^6 - 713742*x^5 + 1813853*x^4 - 892260*x^3 + 1346246*x^2 - 1607848*x + 1025807, 1)
 

Normalized defining polynomial

\( x^{20} + 29 x^{18} - 19 x^{17} + 363 x^{16} - 516 x^{15} + 2749 x^{14} - 6390 x^{13} + 16099 x^{12} - 42411 x^{11} + 81481 x^{10} - 182224 x^{9} + 268289 x^{8} - 501531 x^{7} + 774271 x^{6} - 713742 x^{5} + 1813853 x^{4} - 892260 x^{3} + 1346246 x^{2} - 1607848 x + 1025807 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(159321702103922125456276290645206016=2^{10}\cdot 3^{2}\cdot 401^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.56$
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:  $2, 3, 401$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{15} - \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{15} + \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{15} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3578633323615667472201324767450941438184143059853767395235717} a^{19} + \frac{286331541367011128194458902379620994201613050340966384279067}{3578633323615667472201324767450941438184143059853767395235717} a^{18} - \frac{85948555173493794202231332206376126428653950588662844339371}{1192877774538555824067108255816980479394714353284589131745239} a^{17} + \frac{24054064187677200434586557984670246532023047830440264816640}{210507842565627498364783809850055378716714297638456905602101} a^{16} + \frac{1716649614936046898027253485748288755389041715282791471358921}{3578633323615667472201324767450941438184143059853767395235717} a^{15} - \frac{1407342923804590552390097380897360667872096581000258331267291}{3578633323615667472201324767450941438184143059853767395235717} a^{14} + \frac{241671643368115822351559726832929769083645265894078506323202}{3578633323615667472201324767450941438184143059853767395235717} a^{13} + \frac{836622481463380749485668128296266934875138375376034812893783}{3578633323615667472201324767450941438184143059853767395235717} a^{12} - \frac{506994417326518280713788854106232873659413235103954994132935}{1192877774538555824067108255816980479394714353284589131745239} a^{11} - \frac{79477882059531559256498167908348836492902007353221349084501}{1192877774538555824067108255816980479394714353284589131745239} a^{10} - \frac{108968828253109301761446859068618864789339232705408005259763}{3578633323615667472201324767450941438184143059853767395235717} a^{9} + \frac{713725797425414088760817654907646484804507179276815314782174}{3578633323615667472201324767450941438184143059853767395235717} a^{8} - \frac{768964999112508160087274231699551997120607969022427396852108}{3578633323615667472201324767450941438184143059853767395235717} a^{7} - \frac{1645074832322227938121888647239734724965018758871519433362060}{3578633323615667472201324767450941438184143059853767395235717} a^{6} + \frac{1728541691189702403940862755451032430346042006348608132966348}{3578633323615667472201324767450941438184143059853767395235717} a^{5} + \frac{938596638242307590308456355285738066481594118947434787896389}{3578633323615667472201324767450941438184143059853767395235717} a^{4} + \frac{93167077993109856827221955351083015437108483584196004290335}{210507842565627498364783809850055378716714297638456905602101} a^{3} - \frac{705774394618946740346093821814698574104744164577810495401778}{3578633323615667472201324767450941438184143059853767395235717} a^{2} + \frac{32276623743994348213985160056631855335130807031337343521177}{1192877774538555824067108255816980479394714353284589131745239} a + \frac{487145581799412020858934787772356860502241806256617301546993}{3578633323615667472201324767450941438184143059853767395235717}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}\times C_{678}$, which has order $2712$ (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:  $9$
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:  \( 795087.603907 \) (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 R R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$2.10.10.5$x^{10} - 9 x^{8} + 50 x^{6} - 50 x^{4} + 45 x^{2} - 5$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
401Data not computed