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Properties

Label 22.0.23863468232...0203.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,3^{11}\cdot 1297^{10}$
Root discriminant $45.03$
Ramified primes $3, 1297$
Class number $241$ (GRH)
Class group $[241]$ (GRH)
Galois group $D_{22}$ (as 22T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, -280, 3281, -596, 13618, -15206, 36905, -34229, 48954, -40589, 43592, -30651, 25292, -14957, 10031, -4849, 2485, -873, 339, -88, 29, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 5*x^21 + 29*x^20 - 88*x^19 + 339*x^18 - 873*x^17 + 2485*x^16 - 4849*x^15 + 10031*x^14 - 14957*x^13 + 25292*x^12 - 30651*x^11 + 43592*x^10 - 40589*x^9 + 48954*x^8 - 34229*x^7 + 36905*x^6 - 15206*x^5 + 13618*x^4 - 596*x^3 + 3281*x^2 - 280*x + 25)
 
gp: K = bnfinit(x^22 - 5*x^21 + 29*x^20 - 88*x^19 + 339*x^18 - 873*x^17 + 2485*x^16 - 4849*x^15 + 10031*x^14 - 14957*x^13 + 25292*x^12 - 30651*x^11 + 43592*x^10 - 40589*x^9 + 48954*x^8 - 34229*x^7 + 36905*x^6 - 15206*x^5 + 13618*x^4 - 596*x^3 + 3281*x^2 - 280*x + 25, 1)
 

Normalized defining polynomial

\( x^{22} - 5 x^{21} + 29 x^{20} - 88 x^{19} + 339 x^{18} - 873 x^{17} + 2485 x^{16} - 4849 x^{15} + 10031 x^{14} - 14957 x^{13} + 25292 x^{12} - 30651 x^{11} + 43592 x^{10} - 40589 x^{9} + 48954 x^{8} - 34229 x^{7} + 36905 x^{6} - 15206 x^{5} + 13618 x^{4} - 596 x^{3} + 3281 x^{2} - 280 x + 25 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 11]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2386346823270991037094903689868490203=-\,3^{11}\cdot 1297^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.03$
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:  $3, 1297$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{10} - \frac{1}{5} a^{4} + \frac{1}{5} a$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{11} - \frac{1}{5} a^{5} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{15} - \frac{2}{5} a^{12} - \frac{1}{5} a^{6} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{16} + \frac{1}{5} a^{10} - \frac{1}{5} a^{7} - \frac{1}{5} a^{4} + \frac{2}{5} a$, $\frac{1}{5} a^{17} + \frac{1}{5} a^{11} - \frac{1}{5} a^{8} - \frac{1}{5} a^{5} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{18} + \frac{1}{5} a^{12} - \frac{1}{5} a^{9} - \frac{1}{5} a^{6} + \frac{2}{5} a^{3}$, $\frac{1}{935} a^{19} + \frac{29}{935} a^{18} + \frac{63}{935} a^{17} - \frac{1}{55} a^{16} + \frac{52}{935} a^{15} - \frac{1}{17} a^{14} + \frac{3}{935} a^{13} - \frac{7}{187} a^{12} + \frac{23}{935} a^{11} - \frac{362}{935} a^{10} + \frac{316}{935} a^{9} - \frac{118}{935} a^{8} - \frac{249}{935} a^{7} - \frac{31}{935} a^{6} - \frac{8}{85} a^{5} + \frac{367}{935} a^{4} - \frac{87}{187} a^{3} - \frac{2}{55} a^{2} + \frac{48}{935} a + \frac{89}{187}$, $\frac{1}{4675} a^{20} - \frac{2}{4675} a^{19} - \frac{8}{425} a^{18} - \frac{4}{187} a^{17} + \frac{392}{4675} a^{16} + \frac{203}{4675} a^{15} + \frac{212}{4675} a^{14} + \frac{59}{4675} a^{13} - \frac{1884}{4675} a^{12} + \frac{47}{4675} a^{11} - \frac{243}{4675} a^{10} - \frac{377}{4675} a^{9} - \frac{1266}{4675} a^{8} - \frac{59}{187} a^{7} - \frac{349}{935} a^{6} + \frac{851}{4675} a^{5} + \frac{2213}{4675} a^{4} + \frac{1857}{4675} a^{3} + \frac{1476}{4675} a^{2} - \frac{59}{935} a + \frac{84}{187}$, $\frac{1}{1819722378663801071209556125} a^{21} + \frac{160672920736930163962289}{1819722378663801071209556125} a^{20} - \frac{7499390421944799331971}{363944475732760214241911225} a^{19} - \frac{1310373645927181719967533}{1819722378663801071209556125} a^{18} - \frac{34540793566345111557178588}{1819722378663801071209556125} a^{17} - \frac{20232360207060071191148904}{363944475732760214241911225} a^{16} - \frac{14176681904650951257419394}{363944475732760214241911225} a^{15} + \frac{497190494702989171424838}{107042492862576533600562125} a^{14} - \frac{34762450624411799866334}{363944475732760214241911225} a^{13} + \frac{730476811614231263372121838}{1819722378663801071209556125} a^{12} + \frac{482991754454111296537889764}{1819722378663801071209556125} a^{11} + \frac{63121085526245497610262203}{363944475732760214241911225} a^{10} - \frac{578994594716180288321972073}{1819722378663801071209556125} a^{9} - \frac{186715195908235696334389326}{1819722378663801071209556125} a^{8} + \frac{67165003976231876628146847}{363944475732760214241911225} a^{7} - \frac{154299571916040190464973614}{1819722378663801071209556125} a^{6} + \frac{333200464636035792187615339}{1819722378663801071209556125} a^{5} + \frac{26237392382728701557408117}{363944475732760214241911225} a^{4} + \frac{336670420015558802155549933}{1819722378663801071209556125} a^{3} - \frac{820030982206104611758496594}{1819722378663801071209556125} a^{2} + \frac{139713028909252760061435909}{363944475732760214241911225} a - \frac{26272970376276790794366984}{72788895146552042848382245}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{241}$, which has order $241$ (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:  $10$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{7004872444344238152046092}{1819722378663801071209556125} a^{21} - \frac{35130899307781461457817487}{1819722378663801071209556125} a^{20} + \frac{40697189694742968622684318}{363944475732760214241911225} a^{19} - \frac{618325726204162391625420311}{1819722378663801071209556125} a^{18} + \frac{2377385256158690286327597104}{1819722378663801071209556125} a^{17} - \frac{1225601164841445508027352953}{363944475732760214241911225} a^{16} + \frac{3483396496987894308658676647}{363944475732760214241911225} a^{15} - \frac{33991957112644517675932919393}{1819722378663801071209556125} a^{14} + \frac{14026237292808905187805933182}{363944475732760214241911225} a^{13} - \frac{104407834024688978531651890879}{1819722378663801071209556125} a^{12} + \frac{175940411164104349787113827713}{1819722378663801071209556125} a^{11} - \frac{42540857679337508781734482254}{363944475732760214241911225} a^{10} + \frac{301367191924638670728596486309}{1819722378663801071209556125} a^{9} - \frac{278928109627158738501653246242}{1819722378663801071209556125} a^{8} + \frac{66904587258420909130020216509}{363944475732760214241911225} a^{7} - \frac{13570750582047481422639900614}{107042492862576533600562125} a^{6} + \frac{14603265555652412141029994289}{107042492862576533600562125} a^{5} - \frac{19519844901127034084393241411}{363944475732760214241911225} a^{4} + \frac{87119000524502414050300205636}{1819722378663801071209556125} a^{3} + \frac{2400348133101033807471255302}{1819722378663801071209556125} a^{2} + \frac{4088637920954072386657573868}{363944475732760214241911225} a + \frac{3059208602713343113056007}{72788895146552042848382245} \) (order $6$)
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:  \( 15484188.8463 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{22}$ (as 22T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 44
The 14 conjugacy class representatives for $D_{22}$
Character table for $D_{22}$

Intermediate fields

\(\Q(\sqrt{-3}) \), 11.11.3670285774226257.1

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

Sibling fields

Galois closure: data not computed
Degree 22 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 $22$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{11}$

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
3Data not computed
1297Data not computed