↑

Properties

Label 22.22.4807060044...8125.1
Degree $22$
Signature $[22, 0]$
Discriminant $5^{11}\cdot 74843^{8}$
Root discriminant $132.40$
Ramified primes $5, 74843$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T13

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5249, -10224, 164785, -256700, -429617, 1206965, -7200, -1817353, 844613, 1246157, -946985, -395648, 471182, 31850, -121744, 13322, 16135, -3703, -924, 345, 4, -11, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 + 4*x^20 + 345*x^19 - 924*x^18 - 3703*x^17 + 16135*x^16 + 13322*x^15 - 121744*x^14 + 31850*x^13 + 471182*x^12 - 395648*x^11 - 946985*x^10 + 1246157*x^9 + 844613*x^8 - 1817353*x^7 - 7200*x^6 + 1206965*x^5 - 429617*x^4 - 256700*x^3 + 164785*x^2 - 10224*x - 5249)
 
gp: K = bnfinit(x^22 - 11*x^21 + 4*x^20 + 345*x^19 - 924*x^18 - 3703*x^17 + 16135*x^16 + 13322*x^15 - 121744*x^14 + 31850*x^13 + 471182*x^12 - 395648*x^11 - 946985*x^10 + 1246157*x^9 + 844613*x^8 - 1817353*x^7 - 7200*x^6 + 1206965*x^5 - 429617*x^4 - 256700*x^3 + 164785*x^2 - 10224*x - 5249, 1)
 

Normalized defining polynomial

\( x^{22} - 11 x^{21} + 4 x^{20} + 345 x^{19} - 924 x^{18} - 3703 x^{17} + 16135 x^{16} + 13322 x^{15} - 121744 x^{14} + 31850 x^{13} + 471182 x^{12} - 395648 x^{11} - 946985 x^{10} + 1246157 x^{9} + 844613 x^{8} - 1817353 x^{7} - 7200 x^{6} + 1206965 x^{5} - 429617 x^{4} - 256700 x^{3} + 164785 x^{2} - 10224 x - 5249 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[22, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(48070600443250243342242248871561405585986328125=5^{11}\cdot 74843^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $132.40$
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:  $5, 74843$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{15} + \frac{1}{3} a^{14} - \frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{14} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{5} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{17} - \frac{1}{9} a^{16} + \frac{4}{9} a^{15} + \frac{1}{9} a^{14} + \frac{2}{9} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{4}{9} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{9} a^{7} - \frac{1}{3} a^{5} - \frac{1}{9} a^{4} + \frac{4}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{3} a + \frac{1}{9}$, $\frac{1}{9} a^{19} + \frac{1}{9} a^{17} - \frac{1}{9} a^{16} + \frac{1}{9} a^{14} + \frac{4}{9} a^{13} - \frac{1}{3} a^{12} - \frac{2}{9} a^{11} + \frac{2}{9} a^{10} + \frac{1}{3} a^{9} + \frac{4}{9} a^{8} - \frac{4}{9} a^{7} + \frac{1}{3} a^{6} - \frac{4}{9} a^{5} - \frac{1}{9} a^{4} + \frac{1}{9} a^{3} + \frac{1}{9} a^{2} + \frac{1}{9} a - \frac{1}{9}$, $\frac{1}{27} a^{20} + \frac{1}{27} a^{19} - \frac{4}{27} a^{17} - \frac{1}{9} a^{15} - \frac{11}{27} a^{14} + \frac{8}{27} a^{13} - \frac{8}{27} a^{12} - \frac{2}{27} a^{10} + \frac{4}{27} a^{9} - \frac{2}{9} a^{8} - \frac{2}{27} a^{7} + \frac{8}{27} a^{6} - \frac{8}{27} a^{5} + \frac{1}{27} a^{4} - \frac{2}{27} a^{3} + \frac{1}{9} a^{2} - \frac{2}{9} a + \frac{10}{27}$, $\frac{1}{273250913026288698270417336634050597} a^{21} + \frac{1403824621319633967171075470320904}{91083637675429566090139112211350199} a^{20} + \frac{7975329450427784685247418182522625}{273250913026288698270417336634050597} a^{19} + \frac{8206357942984471306813794647774045}{273250913026288698270417336634050597} a^{18} - \frac{24825163628494820682632873308134431}{273250913026288698270417336634050597} a^{17} + \frac{390529812208246263365527451666372}{91083637675429566090139112211350199} a^{16} - \frac{112886946685960116736552054725746585}{273250913026288698270417336634050597} a^{15} + \frac{37245927248566509015668178911806963}{273250913026288698270417336634050597} a^{14} - \frac{2440704003661189728675218613870511}{273250913026288698270417336634050597} a^{13} - \frac{48903108068477369068317541837460965}{273250913026288698270417336634050597} a^{12} + \frac{36818859008466882846602966224109899}{273250913026288698270417336634050597} a^{11} - \frac{42886490758966701899026630408938370}{91083637675429566090139112211350199} a^{10} - \frac{56289261045241947399293861054538349}{273250913026288698270417336634050597} a^{9} + \frac{108117290935389963420550337761246354}{273250913026288698270417336634050597} a^{8} - \frac{48776652287756371334047379347157795}{273250913026288698270417336634050597} a^{7} - \frac{49635518571276525122417943351178369}{273250913026288698270417336634050597} a^{6} - \frac{4312355679922189872816198297596360}{91083637675429566090139112211350199} a^{5} - \frac{9394292189961123051149633681605118}{91083637675429566090139112211350199} a^{4} + \frac{52925490598501651830881799069976484}{273250913026288698270417336634050597} a^{3} + \frac{40326522534285580297161857810809955}{91083637675429566090139112211350199} a^{2} + \frac{112599905193015235165869145692903154}{273250913026288698270417336634050597} a + \frac{118115284736194600783904886605587730}{273250913026288698270417336634050597}$

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:  $21$
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:  \( 23316714717000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T13:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1320
The 16 conjugacy class representatives for t22n13
Character table for t22n13

Intermediate fields

\(\Q(\sqrt{5}) \), 11.11.31376518243389673201.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 24 sibling: data not computed
Arithmetically equvalently 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 $22$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ $22$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ $22$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $22$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
74843Data not computed