↑

Properties

Label 20.6.18079489015...9712.2
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{10}\cdot 11^{16}\cdot 727^{3}$
Root discriminant $25.87$
Ramified primes $2, 11, 727$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T747

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-43, -54, 256, 68, -1472, -895, 5614, 796, -4251, 1546, 830, -1121, 297, 132, -63, -5, -26, 20, 1, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + x^18 + 20*x^17 - 26*x^16 - 5*x^15 - 63*x^14 + 132*x^13 + 297*x^12 - 1121*x^11 + 830*x^10 + 1546*x^9 - 4251*x^8 + 796*x^7 + 5614*x^6 - 895*x^5 - 1472*x^4 + 68*x^3 + 256*x^2 - 54*x - 43)
 
gp: K = bnfinit(x^20 - 4*x^19 + x^18 + 20*x^17 - 26*x^16 - 5*x^15 - 63*x^14 + 132*x^13 + 297*x^12 - 1121*x^11 + 830*x^10 + 1546*x^9 - 4251*x^8 + 796*x^7 + 5614*x^6 - 895*x^5 - 1472*x^4 + 68*x^3 + 256*x^2 - 54*x - 43, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + x^{18} + 20 x^{17} - 26 x^{16} - 5 x^{15} - 63 x^{14} + 132 x^{13} + 297 x^{12} - 1121 x^{11} + 830 x^{10} + 1546 x^{9} - 4251 x^{8} + 796 x^{7} + 5614 x^{6} - 895 x^{5} - 1472 x^{4} + 68 x^{3} + 256 x^{2} - 54 x - 43 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-18079489015266793067734899712=-\,2^{10}\cdot 11^{16}\cdot 727^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.87$
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:  $2, 11, 727$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{45824096678803391293473671709519649} a^{19} + \frac{10531084420664248412881312352175596}{45824096678803391293473671709519649} a^{18} - \frac{14933805886394709487802969724129341}{45824096678803391293473671709519649} a^{17} + \frac{5986431677614869607506747120318295}{45824096678803391293473671709519649} a^{16} - \frac{890173893238226413500462450266004}{45824096678803391293473671709519649} a^{15} + \frac{8998603600894589927617668037143465}{45824096678803391293473671709519649} a^{14} + \frac{4526032294341499309814416272475659}{45824096678803391293473671709519649} a^{13} - \frac{13066606087056420779648915119523286}{45824096678803391293473671709519649} a^{12} + \frac{3262304671618675917797207077727843}{45824096678803391293473671709519649} a^{11} + \frac{10088791825640737705997793950923987}{45824096678803391293473671709519649} a^{10} + \frac{12521095558670215381747712719163418}{45824096678803391293473671709519649} a^{9} + \frac{3624817914542675178480034648575664}{45824096678803391293473671709519649} a^{8} - \frac{21082153043662795113356138855081869}{45824096678803391293473671709519649} a^{7} - \frac{5780566426159972972313052265325680}{45824096678803391293473671709519649} a^{6} - \frac{1796322799469136128168684836315470}{45824096678803391293473671709519649} a^{5} + \frac{22265983172725340644466805302030442}{45824096678803391293473671709519649} a^{4} - \frac{8430431926639659361149476560944984}{45824096678803391293473671709519649} a^{3} + \frac{12855106391121596115284964050861816}{45824096678803391293473671709519649} a^{2} - \frac{20497325425778195208488209479759228}{45824096678803391293473671709519649} a - \frac{703303217279513598040076254904635}{45824096678803391293473671709519649}$

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:  $12$
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:  \( 2096812.37539 \) (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

\(\Q(\zeta_{11})^+\), 10.8.155838906487.1

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 $20$ $20$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $20$

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$2.10.10.4$x^{10} - 5 x^{8} + 14 x^{6} - 22 x^{4} + 17 x^{2} - 37$$2$$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}$
11Data not computed
727Data not computed