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Properties

Label 21.3.714...224.1
Degree $21$
Signature $[3, 9]$
Discriminant $-7.147\times 10^{27}$
Root discriminant $21.20$
Ramified primes $2, 3$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $SO(3,7)$ (as 21T20)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^20 - 4*x^19 + 16*x^18 - 9*x^17 - 33*x^16 + 52*x^15 + 12*x^14 - 136*x^13 + 96*x^12 + 180*x^11 - 260*x^10 - 144*x^9 + 312*x^8 + 44*x^7 - 220*x^6 + 15*x^5 + 59*x^4 - 16*x^3 + 36*x^2 - 31*x + 1)
 
gp: K = bnfinit(x^21 - 3*x^20 - 4*x^19 + 16*x^18 - 9*x^17 - 33*x^16 + 52*x^15 + 12*x^14 - 136*x^13 + 96*x^12 + 180*x^11 - 260*x^10 - 144*x^9 + 312*x^8 + 44*x^7 - 220*x^6 + 15*x^5 + 59*x^4 - 16*x^3 + 36*x^2 - 31*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -31, 36, -16, 59, 15, -220, 44, 312, -144, -260, 180, 96, -136, 12, 52, -33, -9, 16, -4, -3, 1]);
 

\(x^{21} - 3 x^{20} - 4 x^{19} + 16 x^{18} - 9 x^{17} - 33 x^{16} + 52 x^{15} + 12 x^{14} - 136 x^{13} + 96 x^{12} + 180 x^{11} - 260 x^{10} - 144 x^{9} + 312 x^{8} + 44 x^{7} - 220 x^{6} + 15 x^{5} + 59 x^{4} - 16 x^{3} + 36 x^{2} - 31 x + 1\)  Toggle raw display

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

Invariants

Degree:  $21$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[3, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-7146646609494406531041460224\)\(\medspace = -\,2^{64}\cdot 3^{18}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $21.20$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 3$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} + \frac{1}{4} a^{4} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} + \frac{1}{4} a^{5} - \frac{1}{4} a$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{12} + \frac{1}{8} a^{10} - \frac{1}{8} a^{8} + \frac{1}{8} a^{6} - \frac{1}{8} a^{4} + \frac{1}{8} a^{2} - \frac{1}{8}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{13} + \frac{1}{8} a^{11} - \frac{1}{8} a^{9} + \frac{1}{8} a^{7} - \frac{1}{8} a^{5} + \frac{1}{8} a^{3} - \frac{1}{8} a$, $\frac{1}{8} a^{16} - \frac{1}{8}$, $\frac{1}{64} a^{17} - \frac{3}{64} a^{16} + \frac{1}{32} a^{15} - \frac{1}{32} a^{14} + \frac{1}{16} a^{13} + \frac{7}{32} a^{11} + \frac{5}{32} a^{10} - \frac{1}{4} a^{9} - \frac{1}{16} a^{8} - \frac{15}{32} a^{7} - \frac{9}{32} a^{6} + \frac{3}{16} a^{5} - \frac{3}{8} a^{4} + \frac{11}{32} a^{3} + \frac{1}{32} a^{2} - \frac{1}{64} a + \frac{15}{64}$, $\frac{1}{64} a^{18} + \frac{1}{64} a^{16} - \frac{1}{16} a^{15} - \frac{1}{32} a^{14} + \frac{1}{16} a^{13} - \frac{1}{32} a^{12} + \frac{3}{16} a^{11} + \frac{7}{32} a^{10} + \frac{1}{16} a^{9} + \frac{3}{32} a^{8} + \frac{3}{16} a^{7} + \frac{11}{32} a^{6} + \frac{1}{16} a^{5} - \frac{1}{32} a^{4} + \frac{7}{16} a^{3} + \frac{5}{64} a^{2} + \frac{1}{16} a + \frac{21}{64}$, $\frac{1}{128} a^{19} - \frac{1}{128} a^{18} - \frac{1}{128} a^{17} - \frac{7}{128} a^{16} + \frac{3}{64} a^{15} + \frac{1}{64} a^{14} + \frac{5}{64} a^{13} - \frac{5}{64} a^{12} + \frac{7}{64} a^{11} - \frac{3}{64} a^{10} - \frac{3}{64} a^{9} - \frac{5}{64} a^{8} - \frac{25}{64} a^{7} - \frac{27}{64} a^{6} + \frac{29}{64} a^{5} + \frac{27}{64} a^{4} + \frac{37}{128} a^{3} + \frac{19}{128} a^{2} - \frac{21}{128} a - \frac{3}{128}$, $\frac{1}{1157298870656} a^{20} - \frac{1048673419}{1157298870656} a^{19} - \frac{779216257}{1157298870656} a^{18} + \frac{2580505063}{1157298870656} a^{17} - \frac{27905486493}{578649435328} a^{16} - \frac{9131520381}{578649435328} a^{15} + \frac{8596105857}{578649435328} a^{14} + \frac{11920676885}{578649435328} a^{13} + \frac{3984972003}{578649435328} a^{12} - \frac{50090446817}{578649435328} a^{11} + \frac{128354582921}{578649435328} a^{10} + \frac{66215511389}{578649435328} a^{9} + \frac{2597510291}{578649435328} a^{8} - \frac{59552884241}{578649435328} a^{7} + \frac{83339962201}{578649435328} a^{6} - \frac{243029498667}{578649435328} a^{5} - \frac{330362936963}{1157298870656} a^{4} - \frac{191820525839}{1157298870656} a^{3} - \frac{513957271741}{1157298870656} a^{2} - \frac{401219358285}{1157298870656} a - \frac{53116372489}{144662358832}$  Toggle raw display

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 3668230.07566 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{3}\cdot(2\pi)^{9}\cdot 3668230.07566 \cdot 1}{2\sqrt{7146646609494406531041460224}}\approx 2.64901379622$ (assuming GRH)

Galois group

$SO(3,7)$ (as 21T20):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 336
The 9 conjugacy class representatives for $SO(3,7)$
Character table for $SO(3,7)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 8 sibling: data not computed
Degree 14 sibling: data not computed
Degree 16 sibling: data not computed
Degree 24 sibling: data not computed
Degree 28 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 ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.4.9.3$x^{4} + 6 x^{2} + 10$$4$$1$$9$$D_{4}$$[2, 3, 7/2]$
2.8.28.65$x^{8} + 12 x^{4} + 60$$8$$1$$28$$D_{8}$$[2, 3, 7/2, 9/2]$
2.8.27.41$x^{8} + 8 x^{5} + 2 x^{4} + 8 x^{2} + 2$$8$$1$$27$$D_{8}$$[2, 3, 7/2, 9/2]$
3Data not computed