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Properties

Label 21.3.13950884215...9488.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,2^{18}\cdot 3^{28}\cdot 7^{17}$
Root discriminant $37.87$
Ramified primes $2, 3, 7$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3\times F_7$ (as 21T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-62, 1104, -2778, 2044, -6660, 8664, -7050, 13119, -11046, 10203, -10410, 7467, -5612, 3921, -2328, 1319, -654, 291, -106, 27, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 6*x^20 + 27*x^19 - 106*x^18 + 291*x^17 - 654*x^16 + 1319*x^15 - 2328*x^14 + 3921*x^13 - 5612*x^12 + 7467*x^11 - 10410*x^10 + 10203*x^9 - 11046*x^8 + 13119*x^7 - 7050*x^6 + 8664*x^5 - 6660*x^4 + 2044*x^3 - 2778*x^2 + 1104*x - 62)
 
gp: K = bnfinit(x^21 - 6*x^20 + 27*x^19 - 106*x^18 + 291*x^17 - 654*x^16 + 1319*x^15 - 2328*x^14 + 3921*x^13 - 5612*x^12 + 7467*x^11 - 10410*x^10 + 10203*x^9 - 11046*x^8 + 13119*x^7 - 7050*x^6 + 8664*x^5 - 6660*x^4 + 2044*x^3 - 2778*x^2 + 1104*x - 62, 1)
 

Normalized defining polynomial

\( x^{21} - 6 x^{20} + 27 x^{19} - 106 x^{18} + 291 x^{17} - 654 x^{16} + 1319 x^{15} - 2328 x^{14} + 3921 x^{13} - 5612 x^{12} + 7467 x^{11} - 10410 x^{10} + 10203 x^{9} - 11046 x^{8} + 13119 x^{7} - 7050 x^{6} + 8664 x^{5} - 6660 x^{4} + 2044 x^{3} - 2778 x^{2} + 1104 x - 62 \)

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:  \(-1395088421598327334260620375359488=-\,2^{18}\cdot 3^{28}\cdot 7^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.87$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{17} - \frac{1}{4} a^{16} - \frac{1}{4} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} + \frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{19} - \frac{1}{4} a^{15} + \frac{1}{4} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{799754352040540524920089763299667252} a^{20} + \frac{44542814308581971741693556010939873}{799754352040540524920089763299667252} a^{19} + \frac{7134989506258150234664138046374493}{799754352040540524920089763299667252} a^{18} - \frac{88623580333908080932456346276095933}{799754352040540524920089763299667252} a^{17} - \frac{98817743944213593478475037771664841}{399877176020270262460044881649833626} a^{16} + \frac{9741275080882403235846678006542236}{199938588010135131230022440824916813} a^{15} + \frac{44863593571832407485365958809106941}{199938588010135131230022440824916813} a^{14} + \frac{154914293921854937315487966714121909}{399877176020270262460044881649833626} a^{13} + \frac{244577867631063556106256501970980541}{799754352040540524920089763299667252} a^{12} + \frac{250180078759827212795282398473479929}{799754352040540524920089763299667252} a^{11} - \frac{138223582682602921503657875128978353}{799754352040540524920089763299667252} a^{10} - \frac{129801815452696858919267458535075175}{799754352040540524920089763299667252} a^{9} - \frac{184408583354828555580924118486796157}{399877176020270262460044881649833626} a^{8} + \frac{76919720020528003501102936257231495}{199938588010135131230022440824916813} a^{7} - \frac{80916631776369500978686945439566691}{399877176020270262460044881649833626} a^{6} - \frac{65015517462320801648207072912790253}{399877176020270262460044881649833626} a^{5} + \frac{104533356539661002257110194699480657}{399877176020270262460044881649833626} a^{4} - \frac{907214398665484486844630156072441}{399877176020270262460044881649833626} a^{3} - \frac{9735683985469462227310910903731161}{399877176020270262460044881649833626} a^{2} + \frac{91743235524488883140753121820595779}{199938588010135131230022440824916813} a + \frac{31894411920991715439524299383161602}{199938588010135131230022440824916813}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 470357525.1874316 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times F_7$ (as 21T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 126
The 21 conjugacy class representatives for $C_3\times F_7$
Character table for $C_3\times F_7$ is not computed

Intermediate fields

3.3.3969.1, 7.1.144027072.1

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

Sibling fields

Degree 21 siblings: data not computed
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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ $21$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ $21$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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$2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
3Data not computed
7Data not computed