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Properties

Label 20.4.33170570354...1216.2
Degree $20$
Signature $[4, 8]$
Discriminant $2^{10}\cdot 11^{16}\cdot 89^{3}$
Root discriminant $18.88$
Ramified primes $2, 11, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T747

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, 9, 43, 10, -127, -74, 183, 0, -185, 132, 104, -145, -12, 51, -14, 0, 9, -6, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 6*x^18 + 9*x^17 - 14*x^15 + 51*x^14 - 12*x^13 - 145*x^12 + 104*x^11 + 132*x^10 - 185*x^9 + 183*x^7 - 74*x^6 - 127*x^5 + 10*x^4 + 43*x^3 + 9*x^2 - 1)
 
gp: K = bnfinit(x^20 - x^19 - 6*x^18 + 9*x^17 - 14*x^15 + 51*x^14 - 12*x^13 - 145*x^12 + 104*x^11 + 132*x^10 - 185*x^9 + 183*x^7 - 74*x^6 - 127*x^5 + 10*x^4 + 43*x^3 + 9*x^2 - 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 6 x^{18} + 9 x^{17} - 14 x^{15} + 51 x^{14} - 12 x^{13} - 145 x^{12} + 104 x^{11} + 132 x^{10} - 185 x^{9} + 183 x^{7} - 74 x^{6} - 127 x^{5} + 10 x^{4} + 43 x^{3} + 9 x^{2} - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(33170570354885225234441216=2^{10}\cdot 11^{16}\cdot 89^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.88$
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:  $2, 11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{43} a^{18} + \frac{18}{43} a^{17} - \frac{19}{43} a^{15} - \frac{17}{43} a^{14} - \frac{16}{43} a^{13} - \frac{2}{43} a^{12} - \frac{6}{43} a^{11} - \frac{17}{43} a^{10} - \frac{9}{43} a^{9} - \frac{3}{43} a^{8} - \frac{13}{43} a^{7} - \frac{13}{43} a^{6} + \frac{4}{43} a^{5} - \frac{16}{43} a^{4} - \frac{12}{43} a^{3} - \frac{2}{43} a^{2} - \frac{5}{43} a - \frac{16}{43}$, $\frac{1}{380241180047755511} a^{19} - \frac{3116734905129971}{380241180047755511} a^{18} - \frac{55488629056436127}{380241180047755511} a^{17} - \frac{120977894506086704}{380241180047755511} a^{16} - \frac{51185549578094483}{380241180047755511} a^{15} + \frac{97829604692960958}{380241180047755511} a^{14} - \frac{16616745172057311}{380241180047755511} a^{13} + \frac{121673644358971423}{380241180047755511} a^{12} - \frac{135675300558008976}{380241180047755511} a^{11} - \frac{16572571431540824}{380241180047755511} a^{10} + \frac{185905663353474072}{380241180047755511} a^{9} - \frac{127835903024525243}{380241180047755511} a^{8} - \frac{131935519310443305}{380241180047755511} a^{7} - \frac{45016723836362154}{380241180047755511} a^{6} - \frac{145765889035562126}{380241180047755511} a^{5} - \frac{24634980080672438}{380241180047755511} a^{4} + \frac{89447139711418534}{380241180047755511} a^{3} - \frac{172518010505936757}{380241180047755511} a^{2} + \frac{174156579971150696}{380241180047755511} a + \frac{127843862326731554}{380241180047755511}$

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:  $11$
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:  \( 61177.2163939 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T747:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n747 are not computed
Character table for t20n747 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.2.19077940409.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $20$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $20$ R $20$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$89$$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.4.3.1$x^{4} - 89$$4$$1$$3$$C_4$$[\ ]_{4}$