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Properties

Label 20.10.3101144877...4448.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{20}\cdot 11^{16}\cdot 23^{5}$
Root discriminant $29.82$
Ramified primes $2, 11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T749

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-109, 26, 1030, -3318, 3613, 2060, -12163, 17316, -11045, -1656, 9966, -9274, 4020, 146, -1344, 844, -233, -12, 33, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 33*x^18 - 12*x^17 - 233*x^16 + 844*x^15 - 1344*x^14 + 146*x^13 + 4020*x^12 - 9274*x^11 + 9966*x^10 - 1656*x^9 - 11045*x^8 + 17316*x^7 - 12163*x^6 + 2060*x^5 + 3613*x^4 - 3318*x^3 + 1030*x^2 + 26*x - 109)
 
gp: K = bnfinit(x^20 - 10*x^19 + 33*x^18 - 12*x^17 - 233*x^16 + 844*x^15 - 1344*x^14 + 146*x^13 + 4020*x^12 - 9274*x^11 + 9966*x^10 - 1656*x^9 - 11045*x^8 + 17316*x^7 - 12163*x^6 + 2060*x^5 + 3613*x^4 - 3318*x^3 + 1030*x^2 + 26*x - 109, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 33 x^{18} - 12 x^{17} - 233 x^{16} + 844 x^{15} - 1344 x^{14} + 146 x^{13} + 4020 x^{12} - 9274 x^{11} + 9966 x^{10} - 1656 x^{9} - 11045 x^{8} + 17316 x^{7} - 12163 x^{6} + 2060 x^{5} + 3613 x^{4} - 3318 x^{3} + 1030 x^{2} + 26 x - 109 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-310114487798903480985555304448=-\,2^{20}\cdot 11^{16}\cdot 23^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.82$
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:  $2, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{7742841937513} a^{18} - \frac{9}{7742841937513} a^{17} + \frac{2201796344625}{7742841937513} a^{16} - \frac{2128686881770}{7742841937513} a^{15} + \frac{499455535632}{7742841937513} a^{14} + \frac{2784463930785}{7742841937513} a^{13} + \frac{2574508776115}{7742841937513} a^{12} - \frac{1386340062031}{7742841937513} a^{11} + \frac{1037538302021}{7742841937513} a^{10} - \frac{228305875530}{7742841937513} a^{9} - \frac{3556447989784}{7742841937513} a^{8} - \frac{370747651475}{7742841937513} a^{7} + \frac{1754863723246}{7742841937513} a^{6} + \frac{820568383765}{7742841937513} a^{5} - \frac{3585797590657}{7742841937513} a^{4} - \frac{147910497490}{7742841937513} a^{3} - \frac{3632784339401}{7742841937513} a^{2} + \frac{3363825891957}{7742841937513} a + \frac{2198014326632}{7742841937513}$, $\frac{1}{12597603832333651} a^{19} + \frac{804}{12597603832333651} a^{18} + \frac{249972738337724}{12597603832333651} a^{17} - \frac{333606949580207}{12597603832333651} a^{16} + \frac{5176492028918218}{12597603832333651} a^{15} - \frac{2610866541230469}{12597603832333651} a^{14} + \frac{5355737617572007}{12597603832333651} a^{13} - \frac{5604695590968458}{12597603832333651} a^{12} + \frac{4162304111192197}{12597603832333651} a^{11} + \frac{5845166225300941}{12597603832333651} a^{10} + \frac{128287394642359}{12597603832333651} a^{9} + \frac{4587822348291691}{12597603832333651} a^{8} - \frac{3822656058494344}{12597603832333651} a^{7} + \frac{126727329880579}{12597603832333651} a^{6} + \frac{5286012935105549}{12597603832333651} a^{5} - \frac{3472885971202567}{12597603832333651} a^{4} - \frac{3584934364867082}{12597603832333651} a^{3} + \frac{5319265347222828}{12597603832333651} a^{2} - \frac{2868829098271739}{12597603832333651} a - \frac{5181572096209884}{12597603832333651}$

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:  $14$
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:  \( 25203903.6334 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T749:

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 t20n749 are not computed
Character table for t20n749 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.116117348402176.2

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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$