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Properties

Label 22.0.36676822763...0000.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,2^{20}\cdot 5^{8}\cdot 11^{23}$
Root discriminant $41.36$
Ramified primes $2, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T14

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![86485, -314545, 577951, -754798, 779031, -684519, 546073, -361944, 159434, -33242, 6358, -15368, 13442, -2046, -5082, 5016, -2255, 363, 187, -154, 55, -11, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 + 55*x^20 - 154*x^19 + 187*x^18 + 363*x^17 - 2255*x^16 + 5016*x^15 - 5082*x^14 - 2046*x^13 + 13442*x^12 - 15368*x^11 + 6358*x^10 - 33242*x^9 + 159434*x^8 - 361944*x^7 + 546073*x^6 - 684519*x^5 + 779031*x^4 - 754798*x^3 + 577951*x^2 - 314545*x + 86485)
 
gp: K = bnfinit(x^22 - 11*x^21 + 55*x^20 - 154*x^19 + 187*x^18 + 363*x^17 - 2255*x^16 + 5016*x^15 - 5082*x^14 - 2046*x^13 + 13442*x^12 - 15368*x^11 + 6358*x^10 - 33242*x^9 + 159434*x^8 - 361944*x^7 + 546073*x^6 - 684519*x^5 + 779031*x^4 - 754798*x^3 + 577951*x^2 - 314545*x + 86485, 1)
 

Normalized defining polynomial

\( x^{22} - 11 x^{21} + 55 x^{20} - 154 x^{19} + 187 x^{18} + 363 x^{17} - 2255 x^{16} + 5016 x^{15} - 5082 x^{14} - 2046 x^{13} + 13442 x^{12} - 15368 x^{11} + 6358 x^{10} - 33242 x^{9} + 159434 x^{8} - 361944 x^{7} + 546073 x^{6} - 684519 x^{5} + 779031 x^{4} - 754798 x^{3} + 577951 x^{2} - 314545 x + 86485 \)

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:  \(-366768227637345227672179097600000000=-\,2^{20}\cdot 5^{8}\cdot 11^{23}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.36$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{22} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{3}{11}$, $\frac{1}{44} a^{12} - \frac{1}{44} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{5}{44} a + \frac{17}{44}$, $\frac{1}{44} a^{13} - \frac{1}{44} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{17}{44} a^{2} + \frac{17}{44}$, $\frac{1}{44} a^{14} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{3}{22} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{44} a^{15} - \frac{1}{4} a^{9} + \frac{1}{4} a^{7} - \frac{3}{22} a^{4} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{88} a^{16} - \frac{1}{88} a^{15} - \frac{1}{88} a^{13} + \frac{1}{88} a^{11} - \frac{1}{4} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{8} + \frac{3}{8} a^{7} - \frac{1}{2} a^{6} + \frac{5}{88} a^{5} - \frac{2}{11} a^{4} - \frac{3}{8} a^{3} - \frac{19}{44} a^{2} - \frac{3}{8} a + \frac{5}{88}$, $\frac{1}{88} a^{17} - \frac{1}{88} a^{15} - \frac{1}{88} a^{14} - \frac{1}{88} a^{13} - \frac{1}{88} a^{12} + \frac{1}{88} a^{11} + \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} + \frac{3}{8} a^{7} + \frac{5}{88} a^{6} + \frac{3}{8} a^{5} + \frac{17}{88} a^{4} + \frac{39}{88} a^{3} + \frac{17}{88} a^{2} - \frac{19}{44} a + \frac{27}{88}$, $\frac{1}{88} a^{18} - \frac{1}{88} a^{14} - \frac{1}{88} a^{12} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} + \frac{1}{8} a^{8} + \frac{2}{11} a^{7} + \frac{3}{8} a^{6} - \frac{1}{2} a^{5} + \frac{3}{8} a^{4} - \frac{2}{11} a^{3} - \frac{1}{4} a^{2} - \frac{19}{44} a - \frac{1}{8}$, $\frac{1}{88} a^{19} - \frac{1}{88} a^{15} - \frac{1}{88} a^{13} + \frac{1}{88} a^{11} - \frac{1}{4} a^{10} + \frac{1}{8} a^{9} + \frac{2}{11} a^{8} - \frac{1}{8} a^{7} - \frac{1}{2} a^{6} - \frac{1}{8} a^{5} - \frac{2}{11} a^{4} - \frac{1}{4} a^{3} - \frac{19}{44} a^{2} + \frac{3}{8} a + \frac{2}{11}$, $\frac{1}{1496} a^{20} - \frac{1}{748} a^{19} + \frac{5}{1496} a^{17} - \frac{3}{748} a^{16} - \frac{3}{374} a^{15} - \frac{2}{187} a^{14} + \frac{3}{374} a^{13} + \frac{1}{374} a^{12} - \frac{3}{748} a^{11} + \frac{13}{68} a^{10} + \frac{35}{187} a^{9} - \frac{26}{187} a^{8} - \frac{5}{34} a^{7} - \frac{167}{374} a^{6} + \frac{163}{374} a^{5} + \frac{23}{88} a^{4} - \frac{21}{187} a^{3} + \frac{63}{748} a^{2} + \frac{603}{1496} a + \frac{205}{748}$, $\frac{1}{581609638563290032302959876216} a^{21} - \frac{49656706592543179684061075}{290804819281645016151479938108} a^{20} + \frac{1208914315014397211201766007}{581609638563290032302959876216} a^{19} + \frac{398783682630531230630444529}{83087091223327147471851410888} a^{18} - \frac{854744944198550978564232319}{290804819281645016151479938108} a^{17} - \frac{1410612895311698612121899235}{581609638563290032302959876216} a^{16} + \frac{953224546408016607028302127}{290804819281645016151479938108} a^{15} + \frac{2320624508078456166889785289}{290804819281645016151479938108} a^{14} + \frac{354651306070444054572240543}{41543545611663573735925705444} a^{13} - \frac{339742629610691034456440896}{72701204820411254037869984527} a^{12} - \frac{1624907185446763187447014717}{72701204820411254037869984527} a^{11} + \frac{9904118718074198491331199172}{72701204820411254037869984527} a^{10} - \frac{28224601385008840466924165697}{145402409640822508075739969054} a^{9} - \frac{992311822860372912029214265}{17106165840096765655969408124} a^{8} - \frac{90078716878677249499044138023}{290804819281645016151479938108} a^{7} + \frac{143522181278614466316515712037}{290804819281645016151479938108} a^{6} - \frac{192805193931676773771144316171}{581609638563290032302959876216} a^{5} + \frac{20254571719992541407672766313}{290804819281645016151479938108} a^{4} + \frac{185522608128046325839891066985}{581609638563290032302959876216} a^{3} + \frac{170854007501456930819294467493}{581609638563290032302959876216} a^{2} + \frac{50257031124118719037594421893}{290804819281645016151479938108} a - \frac{235636588824046863186529}{235374196100076905019409096}$

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:  \( -1 \) (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:  \( 3435709897.46 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T14:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1320
The 13 conjugacy class representatives for t22n14
Character table for t22n14

Intermediate fields

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

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

Sibling fields

Degree 12 sibling: data not computed
Degree 24 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 R ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ R ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.8.8.11$x^{8} + 20 x^{2} + 4$$4$$2$$8$$S_4$$[4/3, 4/3]_{3}^{2}$
2.12.12.28$x^{12} - x^{10} + 2 x^{8} - x^{6} - 2 x^{4} + 3 x^{2} + 1$$6$$2$$12$$S_4$$[4/3, 4/3]_{3}^{2}$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11Data not computed