↑

Properties

Label 21.19.8848772031...2643.1
Degree $21$
Signature $[19, 1]$
Discriminant $-\,421\cdot 647\cdot 991\cdot 2843\cdot 53357069\cdot 21609969352347848780355937$
Root discriminant $138.14$
Ramified primes $421, 647, 991, 2843, 53357069, 21609969352347848780355937$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T164

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, -22, 120, 17, -784, 621, 2029, -2423, -2566, 3942, 1633, -3411, -409, 1678, -71, -467, 66, 68, -14, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 4*x^20 - 14*x^19 + 68*x^18 + 66*x^17 - 467*x^16 - 71*x^15 + 1678*x^14 - 409*x^13 - 3411*x^12 + 1633*x^11 + 3942*x^10 - 2566*x^9 - 2423*x^8 + 2029*x^7 + 621*x^6 - 784*x^5 + 17*x^4 + 120*x^3 - 22*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^21 - 4*x^20 - 14*x^19 + 68*x^18 + 66*x^17 - 467*x^16 - 71*x^15 + 1678*x^14 - 409*x^13 - 3411*x^12 + 1633*x^11 + 3942*x^10 - 2566*x^9 - 2423*x^8 + 2029*x^7 + 621*x^6 - 784*x^5 + 17*x^4 + 120*x^3 - 22*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{21} - 4 x^{20} - 14 x^{19} + 68 x^{18} + 66 x^{17} - 467 x^{16} - 71 x^{15} + 1678 x^{14} - 409 x^{13} - 3411 x^{12} + 1633 x^{11} + 3942 x^{10} - 2566 x^{9} - 2423 x^{8} + 2029 x^{7} + 621 x^{6} - 784 x^{5} + 17 x^{4} + 120 x^{3} - 22 x^{2} - 4 x + 1 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[19, 1]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-884877203125648650803983313687121298332172643=-\,421\cdot 647\cdot 991\cdot 2843\cdot 53357069\cdot 21609969352347848780355937\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $138.14$
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:  $421, 647, 991, 2843, 53357069, 21609969352347848780355937$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$

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:  $19$
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:  \( 4501728724140000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T164:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 51090942171709440000
The 792 conjugacy class representatives for S21 are not computed
Character table for S21 is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 42 sibling: 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 ${\href{/LocalNumberField/2.11.0.1}{11} }{,}\,{\href{/LocalNumberField/2.10.0.1}{10} }$ $21$ ${\href{/LocalNumberField/5.14.0.1}{14} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.13.0.1}{13} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.13.0.1}{13} }{,}\,{\href{/LocalNumberField/13.7.0.1}{7} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $19{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.14.0.1}{14} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.11.0.1}{11} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ $16{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }$ ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.11.0.1}{11} }{,}\,{\href{/LocalNumberField/43.10.0.1}{10} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.7.0.1}{7} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ $18{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ $17{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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
421Data not computed
647Data not computed
991Data not computed
2843Data not computed
53357069Data not computed
21609969352347848780355937Data not computed