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Properties

Label 21.3.37403115657...3088.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,2^{14}\cdot 7^{15}\cdot 37^{10}$
Root discriminant $35.57$
Ramified primes $2, 7, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3\times F_7$ (as 21T15)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-290, 4522, -14282, 22177, -8523, -19343, 24604, 2995, -22537, 3557, 12477, -1574, -5106, 243, 1527, -9, -293, -27, 53, 1, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 6*x^20 + x^19 + 53*x^18 - 27*x^17 - 293*x^16 - 9*x^15 + 1527*x^14 + 243*x^13 - 5106*x^12 - 1574*x^11 + 12477*x^10 + 3557*x^9 - 22537*x^8 + 2995*x^7 + 24604*x^6 - 19343*x^5 - 8523*x^4 + 22177*x^3 - 14282*x^2 + 4522*x - 290)
 
gp: K = bnfinit(x^21 - 6*x^20 + x^19 + 53*x^18 - 27*x^17 - 293*x^16 - 9*x^15 + 1527*x^14 + 243*x^13 - 5106*x^12 - 1574*x^11 + 12477*x^10 + 3557*x^9 - 22537*x^8 + 2995*x^7 + 24604*x^6 - 19343*x^5 - 8523*x^4 + 22177*x^3 - 14282*x^2 + 4522*x - 290, 1)
 

Normalized defining polynomial

\( x^{21} - 6 x^{20} + x^{19} + 53 x^{18} - 27 x^{17} - 293 x^{16} - 9 x^{15} + 1527 x^{14} + 243 x^{13} - 5106 x^{12} - 1574 x^{11} + 12477 x^{10} + 3557 x^{9} - 22537 x^{8} + 2995 x^{7} + 24604 x^{6} - 19343 x^{5} - 8523 x^{4} + 22177 x^{3} - 14282 x^{2} + 4522 x - 290 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-374031156573051227998849803993088=-\,2^{14}\cdot 7^{15}\cdot 37^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.57$
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:  $2, 7, 37$
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}{796148116583083764535929184184402795775925} a^{20} - \frac{259292239729698288732324997129562977170203}{796148116583083764535929184184402795775925} a^{19} + \frac{150416872374995671637664918394436226711892}{796148116583083764535929184184402795775925} a^{18} - \frac{363987482340742186689364422134934742992396}{796148116583083764535929184184402795775925} a^{17} + \frac{33350122801515275735070383387383113815267}{159229623316616752907185836836880559155185} a^{16} + \frac{338280772537897054706960533681735783477712}{796148116583083764535929184184402795775925} a^{15} + \frac{290611127300889576100936145221925739089527}{796148116583083764535929184184402795775925} a^{14} - \frac{178173003328196749030556127475210222333967}{796148116583083764535929184184402795775925} a^{13} - \frac{352358698790741133149863645707755344052833}{796148116583083764535929184184402795775925} a^{12} - \frac{30792908495718744686693444825638085569896}{159229623316616752907185836836880559155185} a^{11} - \frac{85402893935499868853114767873089746288639}{796148116583083764535929184184402795775925} a^{10} + \frac{58038902839255511676666977970318858214832}{159229623316616752907185836836880559155185} a^{9} - \frac{57416505850450502178457428209919961662263}{796148116583083764535929184184402795775925} a^{8} + \frac{104109441349500928478368843322983084844999}{796148116583083764535929184184402795775925} a^{7} - \frac{47307510827809937105810172528344005536633}{796148116583083764535929184184402795775925} a^{6} - \frac{42413419211819386053887585677154594011184}{159229623316616752907185836836880559155185} a^{5} - \frac{103830121447669397017342045532153056827753}{796148116583083764535929184184402795775925} a^{4} - \frac{94078744414051075273123226938051346191782}{796148116583083764535929184184402795775925} a^{3} + \frac{114376764532565483851245697411469011366356}{796148116583083764535929184184402795775925} a^{2} - \frac{359270850951602533686579325125281822952189}{796148116583083764535929184184402795775925} a - \frac{1302782140151058893110896761122492700156}{5490676666090232858868477132306226177765}$

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:  $11$
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:  \( 169090528.679 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times F_7$ (as 21T15):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 252
The 21 conjugacy class representatives for $S_3\times F_7$
Character table for $S_3\times F_7$ is not computed

Intermediate fields

3.3.148.1, 7.1.851324971.1

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

Sibling fields

Degree 42 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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.14.0.1}{14} }{,}\,{\href{/LocalNumberField/13.7.0.1}{7} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
7Data not computed
37Data not computed