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Properties

Label 22.22.1086355631...3824.1
Degree $22$
Signature $[22, 0]$
Discriminant $2^{32}\cdot 7^{10}\cdot 11^{23}$
Root discriminant $81.42$
Ramified primes $2, 7, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times F_{11}$ (as 22T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-35, 88, 4389, 2904, -56419, -7920, 227469, 4048, -424182, 1144, 417890, -776, -230406, 0, 72050, 0, -12551, 0, 1177, 0, -55, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 55*x^20 + 1177*x^18 - 12551*x^16 + 72050*x^14 - 230406*x^12 - 776*x^11 + 417890*x^10 + 1144*x^9 - 424182*x^8 + 4048*x^7 + 227469*x^6 - 7920*x^5 - 56419*x^4 + 2904*x^3 + 4389*x^2 + 88*x - 35)
 
gp: K = bnfinit(x^22 - 55*x^20 + 1177*x^18 - 12551*x^16 + 72050*x^14 - 230406*x^12 - 776*x^11 + 417890*x^10 + 1144*x^9 - 424182*x^8 + 4048*x^7 + 227469*x^6 - 7920*x^5 - 56419*x^4 + 2904*x^3 + 4389*x^2 + 88*x - 35, 1)
 

Normalized defining polynomial

\( x^{22} - 55 x^{20} + 1177 x^{18} - 12551 x^{16} + 72050 x^{14} - 230406 x^{12} - 776 x^{11} + 417890 x^{10} + 1144 x^{9} - 424182 x^{8} + 4048 x^{7} + 227469 x^{6} - 7920 x^{5} - 56419 x^{4} + 2904 x^{3} + 4389 x^{2} + 88 x - 35 \)

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:  \(1086355631527929051985063244906158383693824=2^{32}\cdot 7^{10}\cdot 11^{23}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.42$
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:  $2, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{3} + \frac{1}{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{4} a^{4} + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{8} - \frac{1}{4} a^{5} + \frac{1}{4}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{12} - \frac{1}{8} a^{10} + \frac{1}{8} a^{8} - \frac{1}{8} a^{6} + \frac{1}{8} a^{4} + \frac{1}{8} a^{2} - \frac{1}{8}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{5} + \frac{1}{8} a^{3} + \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{8} a^{16} - \frac{1}{4} a^{8} + \frac{1}{8}$, $\frac{1}{8} a^{17} - \frac{1}{4} a^{8} - \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{16} a^{18} - \frac{1}{16} a^{16} - \frac{1}{8} a^{10} + \frac{1}{8} a^{8} + \frac{1}{16} a^{2} - \frac{1}{16}$, $\frac{1}{16} a^{19} - \frac{1}{16} a^{17} - \frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{4} a^{8} + \frac{1}{16} a^{3} + \frac{3}{16} a + \frac{1}{4}$, $\frac{1}{16} a^{20} - \frac{1}{16} a^{16} - \frac{1}{8} a^{12} + \frac{1}{8} a^{8} + \frac{1}{16} a^{4} - \frac{1}{16}$, $\frac{1}{362752781074815285103497507266512} a^{21} + \frac{5260344439514385631294533811793}{181376390537407642551748753633256} a^{20} + \frac{2346114551887882737576390051079}{181376390537407642551748753633256} a^{19} - \frac{267682489220214029215049069109}{22672048817175955318968594204157} a^{18} + \frac{9665057180275298029099780269125}{362752781074815285103497507266512} a^{17} - \frac{7921537302335215226708258661975}{181376390537407642551748753633256} a^{16} - \frac{88960797470681638232165362429}{181376390537407642551748753633256} a^{15} - \frac{1114363696083449194718824702941}{181376390537407642551748753633256} a^{14} - \frac{904852027232531929396712916077}{45344097634351910637937188408314} a^{13} - \frac{9409423998362939962178159156487}{181376390537407642551748753633256} a^{12} - \frac{9422982190836416499059351181663}{181376390537407642551748753633256} a^{11} + \frac{7801629569364155601846077217669}{181376390537407642551748753633256} a^{10} - \frac{4290412862764970675978068709199}{45344097634351910637937188408314} a^{9} - \frac{36651458845201338959610723896897}{181376390537407642551748753633256} a^{8} + \frac{36523678037816889100029432670653}{181376390537407642551748753633256} a^{7} + \frac{18141690667111071464681111333397}{181376390537407642551748753633256} a^{6} - \frac{41286015249594634864262847620273}{362752781074815285103497507266512} a^{5} - \frac{17948800679363766074384516306143}{90688195268703821275874376816628} a^{4} - \frac{6584938028286202871809897186648}{22672048817175955318968594204157} a^{3} + \frac{53734986527056426043540407402507}{181376390537407642551748753633256} a^{2} - \frac{113327083269936290781878515177301}{362752781074815285103497507266512} a + \frac{18489615130404569297557466310885}{45344097634351910637937188408314}$

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:  \( 430410424025000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_{11}$ (as 22T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 220
The 22 conjugacy class representatives for $C_2\times F_{11}$
Character table for $C_2\times F_{11}$ is not computed

Intermediate fields

\(\Q(\sqrt{11}) \), 11.11.4910318845910094848.1

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

Sibling fields

Degree 22 sibling: data not computed
Degree 44 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
2Data not computed
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11Data not computed