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Properties

Label 21.21.5906895800...1232.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{18}\cdot 7^{35}\cdot 29^{6}$
Root discriminant $121.44$
Ramified primes $2, 7, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T31

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-87464, -1271592, -5851678, -7470603, 8406636, 17693074, -4453414, -15314839, 1133986, 7005922, -147546, -1896692, 9338, 317128, -226, -32900, 0, 2051, 0, -70, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 70*x^19 + 2051*x^17 - 32900*x^15 - 226*x^14 + 317128*x^13 + 9338*x^12 - 1896692*x^11 - 147546*x^10 + 7005922*x^9 + 1133986*x^8 - 15314839*x^7 - 4453414*x^6 + 17693074*x^5 + 8406636*x^4 - 7470603*x^3 - 5851678*x^2 - 1271592*x - 87464)
 
gp: K = bnfinit(x^21 - 70*x^19 + 2051*x^17 - 32900*x^15 - 226*x^14 + 317128*x^13 + 9338*x^12 - 1896692*x^11 - 147546*x^10 + 7005922*x^9 + 1133986*x^8 - 15314839*x^7 - 4453414*x^6 + 17693074*x^5 + 8406636*x^4 - 7470603*x^3 - 5851678*x^2 - 1271592*x - 87464, 1)
 

Normalized defining polynomial

\( x^{21} - 70 x^{19} + 2051 x^{17} - 32900 x^{15} - 226 x^{14} + 317128 x^{13} + 9338 x^{12} - 1896692 x^{11} - 147546 x^{10} + 7005922 x^{9} + 1133986 x^{8} - 15314839 x^{7} - 4453414 x^{6} + 17693074 x^{5} + 8406636 x^{4} - 7470603 x^{3} - 5851678 x^{2} - 1271592 x - 87464 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[21, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(59068958006402020727391274026572452122591232=2^{18}\cdot 7^{35}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $121.44$
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:  $2, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{3}$, $\frac{1}{11252} a^{18} + \frac{41}{194} a^{17} - \frac{621}{11252} a^{16} + \frac{21}{194} a^{15} - \frac{2295}{5626} a^{14} - \frac{14}{97} a^{13} + \frac{1559}{5626} a^{12} + \frac{9}{58} a^{11} - \frac{1139}{5626} a^{10} - \frac{1}{97} a^{9} + \frac{7}{58} a^{8} - \frac{1041}{5626} a^{7} - \frac{598}{2813} a^{6} - \frac{8}{29} a^{5} + \frac{1337}{11252} a^{4} + \frac{20}{97} a^{3} + \frac{71}{388} a^{2} + \frac{6}{97} a - \frac{33}{97}$, $\frac{1}{461332} a^{19} - \frac{17}{461332} a^{18} + \frac{59467}{461332} a^{17} - \frac{36133}{461332} a^{16} + \frac{20646}{115333} a^{15} + \frac{106005}{230666} a^{14} + \frac{27833}{230666} a^{13} - \frac{4260}{115333} a^{12} + \frac{22953}{115333} a^{11} + \frac{21741}{230666} a^{10} - \frac{29191}{230666} a^{9} + \frac{19025}{115333} a^{8} + \frac{959}{7954} a^{7} + \frac{19308}{115333} a^{6} + \frac{222885}{461332} a^{5} + \frac{63285}{461332} a^{4} - \frac{51}{15908} a^{3} + \frac{117}{15908} a^{2} + \frac{632}{3977} a + \frac{1823}{3977}$, $\frac{1}{32652737439137667485237601477179024588} a^{20} - \frac{7328955088722833868766542886373}{8163184359784416871309400369294756147} a^{19} + \frac{1067424511596175351561570953106097}{32652737439137667485237601477179024588} a^{18} + \frac{91440843791621916430819059587843416}{8163184359784416871309400369294756147} a^{17} - \frac{282140575643472924185516967348189283}{16326368719568833742618800738589512294} a^{16} + \frac{1854062265877624928386628959430745992}{8163184359784416871309400369294756147} a^{15} + \frac{44064685149814276896468078038565925}{398204115111434969332165871672914934} a^{14} - \frac{3244285714153790259168489855077142217}{16326368719568833742618800738589512294} a^{13} - \frac{6466992949954077126278068581482620377}{16326368719568833742618800738589512294} a^{12} + \frac{258015805515781332446605607481309256}{8163184359784416871309400369294756147} a^{11} - \frac{7404712516843536898395698560020572511}{16326368719568833742618800738589512294} a^{10} - \frac{1431271761699897351324550871332129895}{16326368719568833742618800738589512294} a^{9} + \frac{4007401885940634971904816753986855267}{8163184359784416871309400369294756147} a^{8} + \frac{2551372704100946802438851586801971902}{8163184359784416871309400369294756147} a^{7} - \frac{9376776436900793717585796147626215927}{32652737439137667485237601477179024588} a^{6} + \frac{6657794773829529335812234216989699165}{16326368719568833742618800738589512294} a^{5} + \frac{11018815509067864395944001409233727041}{32652737439137667485237601477179024588} a^{4} + \frac{256683222655409871927740824987270181}{562978231709270129055820715123776286} a^{3} + \frac{89297638214837146649499940757867835}{281489115854635064527910357561888143} a^{2} + \frac{13158816401410229624154697998662840}{281489115854635064527910357561888143} a + \frac{4240506714407222042248875021449564}{281489115854635064527910357561888143}$

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:  $20$
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:  \( 2949939511390000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T31:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2058
The 71 conjugacy class representatives for t21n31 are not computed
Character table for t21n31 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\)

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 $21$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ R $21$ $21$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{7}$ $21$ $21$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
7Data not computed
$29$29.7.6.5$x^{7} + 58$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.0.1$x^{7} - x + 3$$1$$7$$0$$C_7$$[\ ]^{7}$
29.7.0.1$x^{7} - x + 3$$1$$7$$0$$C_7$$[\ ]^{7}$