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Properties

Label 22.0.131...371.1
Degree $22$
Signature $[0, 11]$
Discriminant $-1.311\times 10^{28}$
Root discriminant $18.97$
Ramified prime $11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $F_{11}$ (as 22T4)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 44*x^18 - 22*x^17 + 55*x^16 + 121*x^15 + 187*x^14 + 11*x^13 + 594*x^12 + 328*x^11 + 836*x^10 + 770*x^9 + 814*x^8 + 528*x^7 + 374*x^6 + 143*x^5 + 11*x^4 - 33*x^3 - 11*x^2 + 1)
 
gp: K = bnfinit(x^22 + 44*x^18 - 22*x^17 + 55*x^16 + 121*x^15 + 187*x^14 + 11*x^13 + 594*x^12 + 328*x^11 + 836*x^10 + 770*x^9 + 814*x^8 + 528*x^7 + 374*x^6 + 143*x^5 + 11*x^4 - 33*x^3 - 11*x^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -11, -33, 11, 143, 374, 528, 814, 770, 836, 328, 594, 11, 187, 121, 55, -22, 44, 0, 0, 0, 1]);
 

\( x^{22} + 44 x^{18} - 22 x^{17} + 55 x^{16} + 121 x^{15} + 187 x^{14} + 11 x^{13} + 594 x^{12} + 328 x^{11} + 836 x^{10} + 770 x^{9} + 814 x^{8} + 528 x^{7} + 374 x^{6} + 143 x^{5} + 11 x^{4} - 33 x^{3} - 11 x^{2} + 1 \)

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

Invariants

Degree:  $22$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 11]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-13109994191499930367061460371\)\(\medspace = -\,11^{27}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $18.97$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $11$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{113970065816792274886719464268} a^{21} + \frac{16632653017427876909074098013}{113970065816792274886719464268} a^{20} + \frac{24534011559695427103983197473}{113970065816792274886719464268} a^{19} - \frac{8912679842265051647602954139}{113970065816792274886719464268} a^{18} - \frac{11497465387367424029509679281}{37990021938930758295573154756} a^{17} - \frac{27697277350382184186149372209}{113970065816792274886719464268} a^{16} - \frac{4130993445106208706555625189}{18995010969465379147786577378} a^{15} + \frac{19393796180526678183918660379}{113970065816792274886719464268} a^{14} + \frac{3812146990546715324894643805}{56985032908396137443359732134} a^{13} - \frac{35928163245366322324057443731}{113970065816792274886719464268} a^{12} + \frac{19371841436395072739319519835}{113970065816792274886719464268} a^{11} - \frac{28993452839782047961915623889}{113970065816792274886719464268} a^{10} - \frac{30475100792091463044359391785}{113970065816792274886719464268} a^{9} - \frac{4634344974055439584794468705}{37990021938930758295573154756} a^{8} - \frac{21883060672984889601168832733}{113970065816792274886719464268} a^{7} - \frac{17735108070800523283337582765}{113970065816792274886719464268} a^{6} + \frac{11159970877478780668745224323}{37990021938930758295573154756} a^{5} + \frac{621189490618127479206437519}{28492516454198068721679866067} a^{4} - \frac{48313980575167983792500458385}{113970065816792274886719464268} a^{3} - \frac{18634245538822845828429057679}{56985032908396137443359732134} a^{2} - \frac{1117306900741761353758598413}{113970065816792274886719464268} a - \frac{24046786530289485165087345829}{113970065816792274886719464268}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $10$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 302171.954912 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{11}\cdot 302171.954912 \cdot 1}{2\sqrt{13109994191499930367061460371}}\approx 0.795062849568$ (assuming GRH)

Galois group

$F_{11}$ (as 22T4):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 110
The 11 conjugacy class representatives for $F_{11}$
Character table for $F_{11}$

Intermediate fields

\(\Q(\sqrt{-11}) \), 11.1.34522712143931.1

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

Sibling fields

Degree 11 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
11Data not computed