↑

Properties

Label 20.12.3879990017...9609.2
Degree $20$
Signature $[12, 4]$
Discriminant $3^{2}\cdot 401^{11}$
Root discriminant $30.16$
Ramified primes $3, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T350

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -27, -85, 143, 34, -95, 106, -382, 524, -694, 871, 1014, -1640, -241, 674, -80, -61, 31, -9, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 9*x^18 + 31*x^17 - 61*x^16 - 80*x^15 + 674*x^14 - 241*x^13 - 1640*x^12 + 1014*x^11 + 871*x^10 - 694*x^9 + 524*x^8 - 382*x^7 + 106*x^6 - 95*x^5 + 34*x^4 + 143*x^3 - 85*x^2 - 27*x - 1)
 
gp: K = bnfinit(x^20 - 2*x^19 - 9*x^18 + 31*x^17 - 61*x^16 - 80*x^15 + 674*x^14 - 241*x^13 - 1640*x^12 + 1014*x^11 + 871*x^10 - 694*x^9 + 524*x^8 - 382*x^7 + 106*x^6 - 95*x^5 + 34*x^4 + 143*x^3 - 85*x^2 - 27*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 9 x^{18} + 31 x^{17} - 61 x^{16} - 80 x^{15} + 674 x^{14} - 241 x^{13} - 1640 x^{12} + 1014 x^{11} + 871 x^{10} - 694 x^{9} + 524 x^{8} - 382 x^{7} + 106 x^{6} - 95 x^{5} + 34 x^{4} + 143 x^{3} - 85 x^{2} - 27 x - 1 \)

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:  \(387999001772721822047119239609=3^{2}\cdot 401^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.16$
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:  $3, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{15} - \frac{1}{6} a^{13} + \frac{1}{3} a^{12} - \frac{1}{6} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{2} a^{8} + \frac{1}{3} a^{7} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{6} a^{16} - \frac{1}{6} a^{14} + \frac{1}{3} a^{13} - \frac{1}{6} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{2} a^{9} + \frac{1}{3} a^{8} - \frac{1}{2} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{17} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{2} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{6}$, $\frac{1}{666} a^{18} - \frac{2}{111} a^{17} + \frac{1}{333} a^{16} - \frac{13}{666} a^{15} - \frac{19}{333} a^{14} - \frac{59}{666} a^{13} - \frac{11}{222} a^{12} - \frac{70}{333} a^{11} - \frac{17}{37} a^{10} + \frac{121}{333} a^{9} + \frac{281}{666} a^{8} + \frac{199}{666} a^{7} + \frac{95}{666} a^{6} - \frac{134}{333} a^{5} + \frac{67}{222} a^{4} - \frac{199}{666} a^{3} + \frac{95}{666} a^{2} + \frac{8}{333} a - \frac{55}{666}$, $\frac{1}{9943111532030575796418582} a^{19} + \frac{482423279253057710440}{4971555766015287898209291} a^{18} + \frac{198352511221432975757608}{4971555766015287898209291} a^{17} - \frac{63531130138327042634681}{1657185255338429299403097} a^{16} + \frac{51133421184764522554642}{4971555766015287898209291} a^{15} - \frac{182190788364068364279946}{1657185255338429299403097} a^{14} - \frac{4219138056357534393039193}{9943111532030575796418582} a^{13} + \frac{4136263632609445181951449}{9943111532030575796418582} a^{12} - \frac{1725467061356569222980362}{4971555766015287898209291} a^{11} + \frac{2013267622791899442456802}{4971555766015287898209291} a^{10} - \frac{786451740334771731871117}{1657185255338429299403097} a^{9} + \frac{2830442504511544176185903}{9943111532030575796418582} a^{8} - \frac{2183217307338889355907151}{4971555766015287898209291} a^{7} - \frac{139952348034631858252402}{552395085112809766467699} a^{6} - \frac{794595186295495730107859}{9943111532030575796418582} a^{5} + \frac{1878543779385471413550587}{9943111532030575796418582} a^{4} + \frac{191974269172650038835695}{1104790170225619532935398} a^{3} - \frac{1733040206170093572247693}{9943111532030575796418582} a^{2} + \frac{1153053044120704679135468}{4971555766015287898209291} a - \frac{1645882560639499319458}{4971555766015287898209291}$

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:  \( 51932024.7408 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T350:

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 t20n350 are not computed
Character table for t20n350 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$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}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
401Data not computed