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Properties

Label 20.12.3396666404...5184.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 11^{16}\cdot 89^{3}$
Root discriminant $26.70$
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![23, -12, -292, 46, 874, -312, -1252, 786, 1083, -850, -524, 356, 185, 36, -146, -84, 74, 24, -15, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 15*x^18 + 24*x^17 + 74*x^16 - 84*x^15 - 146*x^14 + 36*x^13 + 185*x^12 + 356*x^11 - 524*x^10 - 850*x^9 + 1083*x^8 + 786*x^7 - 1252*x^6 - 312*x^5 + 874*x^4 + 46*x^3 - 292*x^2 - 12*x + 23)
 
gp: K = bnfinit(x^20 - 2*x^19 - 15*x^18 + 24*x^17 + 74*x^16 - 84*x^15 - 146*x^14 + 36*x^13 + 185*x^12 + 356*x^11 - 524*x^10 - 850*x^9 + 1083*x^8 + 786*x^7 - 1252*x^6 - 312*x^5 + 874*x^4 + 46*x^3 - 292*x^2 - 12*x + 23, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 15 x^{18} + 24 x^{17} + 74 x^{16} - 84 x^{15} - 146 x^{14} + 36 x^{13} + 185 x^{12} + 356 x^{11} - 524 x^{10} - 850 x^{9} + 1083 x^{8} + 786 x^{7} - 1252 x^{6} - 312 x^{5} + 874 x^{4} + 46 x^{3} - 292 x^{2} - 12 x + 23 \)

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:  \(33966664043402470640067805184=2^{20}\cdot 11^{16}\cdot 89^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.70$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{43} a^{18} - \frac{6}{43} a^{17} - \frac{9}{43} a^{16} - \frac{4}{43} a^{15} - \frac{6}{43} a^{14} + \frac{12}{43} a^{13} - \frac{8}{43} a^{11} + \frac{2}{43} a^{10} + \frac{19}{43} a^{9} + \frac{9}{43} a^{8} + \frac{19}{43} a^{7} - \frac{15}{43} a^{6} - \frac{12}{43} a^{5} + \frac{12}{43} a^{4} - \frac{15}{43} a^{3} - \frac{13}{43} a^{2} - \frac{19}{43} a + \frac{18}{43}$, $\frac{1}{2584589586236719201545569} a^{19} + \frac{13808869479029650785652}{2584589586236719201545569} a^{18} + \frac{839499080046373923976023}{2584589586236719201545569} a^{17} - \frac{1037818320685081686655015}{2584589586236719201545569} a^{16} - \frac{181872537119655457908878}{2584589586236719201545569} a^{15} - \frac{886515267844338934567284}{2584589586236719201545569} a^{14} + \frac{1231027873648686845431275}{2584589586236719201545569} a^{13} - \frac{803046857487691669176391}{2584589586236719201545569} a^{12} + \frac{1042600574094527368996432}{2584589586236719201545569} a^{11} - \frac{1103466803222178960195911}{2584589586236719201545569} a^{10} - \frac{1198573489056160435935407}{2584589586236719201545569} a^{9} + \frac{1189226423916094220832977}{2584589586236719201545569} a^{8} - \frac{839433924692796788817145}{2584589586236719201545569} a^{7} + \frac{1412311158692591171927}{60106734563644632594083} a^{6} + \frac{865461713858502904869826}{2584589586236719201545569} a^{5} - \frac{509643217872614883949352}{2584589586236719201545569} a^{4} + \frac{400821780208801722012516}{2584589586236719201545569} a^{3} + \frac{22674793787844120366514}{60106734563644632594083} a^{2} + \frac{320286155804973021585452}{2584589586236719201545569} a + \frac{85356426784788706377668}{2584589586236719201545569}$

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:  \( 10654670.1691 \) (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

\(\Q(\zeta_{11})^+\), 10.10.19535810978816.1

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 $20$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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
2Data not computed
$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}$