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Properties

Label 22.0.22378993818...5627.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,3^{11}\cdot 1831^{8}$
Root discriminant $26.61$
Ramified primes $3, 1831$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T13

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 7, -12, 35, -58, 41, -35, -19, 123, -154, 144, 42, -127, 146, -57, -4, 38, -22, 8, 3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 + 3*x^20 + 8*x^19 - 22*x^18 + 38*x^17 - 4*x^16 - 57*x^15 + 146*x^14 - 127*x^13 + 42*x^12 + 144*x^11 - 154*x^10 + 123*x^9 - 19*x^8 - 35*x^7 + 41*x^6 - 58*x^5 + 35*x^4 - 12*x^3 + 7*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^22 - 2*x^21 + 3*x^20 + 8*x^19 - 22*x^18 + 38*x^17 - 4*x^16 - 57*x^15 + 146*x^14 - 127*x^13 + 42*x^12 + 144*x^11 - 154*x^10 + 123*x^9 - 19*x^8 - 35*x^7 + 41*x^6 - 58*x^5 + 35*x^4 - 12*x^3 + 7*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{22} - 2 x^{21} + 3 x^{20} + 8 x^{19} - 22 x^{18} + 38 x^{17} - 4 x^{16} - 57 x^{15} + 146 x^{14} - 127 x^{13} + 42 x^{12} + 144 x^{11} - 154 x^{10} + 123 x^{9} - 19 x^{8} - 35 x^{7} + 41 x^{6} - 58 x^{5} + 35 x^{4} - 12 x^{3} + 7 x^{2} - 2 x + 1 \)

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:  \(-22378993818717495955548767565627=-\,3^{11}\cdot 1831^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.61$
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:  $3, 1831$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{839} a^{20} + \frac{405}{839} a^{19} + \frac{28}{839} a^{18} - \frac{69}{839} a^{17} + \frac{241}{839} a^{16} + \frac{46}{839} a^{15} + \frac{149}{839} a^{14} + \frac{122}{839} a^{13} + \frac{300}{839} a^{12} + \frac{133}{839} a^{11} - \frac{253}{839} a^{10} + \frac{354}{839} a^{9} - \frac{76}{839} a^{8} - \frac{124}{839} a^{7} - \frac{18}{839} a^{6} + \frac{268}{839} a^{5} - \frac{77}{839} a^{4} - \frac{279}{839} a^{3} + \frac{242}{839} a^{2} + \frac{75}{839} a - \frac{149}{839}$, $\frac{1}{153896253801989} a^{21} + \frac{87588763607}{153896253801989} a^{20} - \frac{22359869945973}{153896253801989} a^{19} + \frac{64017950780500}{153896253801989} a^{18} - \frac{36953227625811}{153896253801989} a^{17} - \frac{6393148815941}{153896253801989} a^{16} + \frac{11874943969860}{153896253801989} a^{15} + \frac{21685095672904}{153896253801989} a^{14} + \frac{36289666758452}{153896253801989} a^{13} + \frac{24144986477432}{153896253801989} a^{12} + \frac{73342327158105}{153896253801989} a^{11} - \frac{75641298112498}{153896253801989} a^{10} - \frac{5883372201045}{153896253801989} a^{9} + \frac{49229852293319}{153896253801989} a^{8} - \frac{53198585229236}{153896253801989} a^{7} + \frac{71443090465014}{153896253801989} a^{6} - \frac{344837388735}{153896253801989} a^{5} - \frac{12979624546535}{153896253801989} a^{4} - \frac{23173787888631}{153896253801989} a^{3} + \frac{41036472978270}{153896253801989} a^{2} - \frac{55934009147780}{153896253801989} a - \frac{19731794366184}{153896253801989}$

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:  $10$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{73270484821}{183428192851} a^{21} + \frac{116409601624}{183428192851} a^{20} - \frac{155178102378}{183428192851} a^{19} - \frac{687197029618}{183428192851} a^{18} + \frac{1386977019282}{183428192851} a^{17} - \frac{2092756379177}{183428192851} a^{16} - \frac{965838247500}{183428192851} a^{15} + \frac{4497579191569}{183428192851} a^{14} - \frac{9074565932759}{183428192851} a^{13} + \frac{4635042678194}{183428192851} a^{12} + \frac{1493279417118}{183428192851} a^{11} - \frac{12677564258950}{183428192851} a^{10} + \frac{7256695913259}{183428192851} a^{9} - \frac{3739012450007}{183428192851} a^{8} - \frac{3466050760805}{183428192851} a^{7} + \frac{3827066469121}{183428192851} a^{6} - \frac{2152777956344}{183428192851} a^{5} + \frac{2351369891221}{183428192851} a^{4} - \frac{542678460350}{183428192851} a^{3} - \frac{459221216873}{183428192851} a^{2} - \frac{73767514697}{183428192851} a + \frac{57360729737}{183428192851} \) (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:  \( 19742415.8494 \) (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

\(\Q(\sqrt{-3}) \), 11.3.11239665258721.2

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 $22$ R $22$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{11}$ $22$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
1831Data not computed