Properties

Label 21.5.267...304.1
Degree $21$
Signature $[5, 8]$
Discriminant $2.673\times 10^{25}$
Root discriminant $16.25$
Ramified primes $2, 317$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $\PSL(2,7)$ (as 21T14)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 5*x^20 + 11*x^19 - 16*x^18 + 25*x^17 - 38*x^16 + 36*x^15 - 25*x^14 + 18*x^13 - 4*x^12 - 3*x^11 - 17*x^10 - 23*x^9 - 12*x^8 + 55*x^7 + 18*x^6 - 46*x^5 - 17*x^4 + 19*x^3 + 9*x^2 - 2*x - 1)
 
gp: K = bnfinit(x^21 - 5*x^20 + 11*x^19 - 16*x^18 + 25*x^17 - 38*x^16 + 36*x^15 - 25*x^14 + 18*x^13 - 4*x^12 - 3*x^11 - 17*x^10 - 23*x^9 - 12*x^8 + 55*x^7 + 18*x^6 - 46*x^5 - 17*x^4 + 19*x^3 + 9*x^2 - 2*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 9, 19, -17, -46, 18, 55, -12, -23, -17, -3, -4, 18, -25, 36, -38, 25, -16, 11, -5, 1]);
 

\( x^{21} - 5 x^{20} + 11 x^{19} - 16 x^{18} + 25 x^{17} - 38 x^{16} + 36 x^{15} - 25 x^{14} + 18 x^{13} - 4 x^{12} - 3 x^{11} - 17 x^{10} - 23 x^{9} - 12 x^{8} + 55 x^{7} + 18 x^{6} - 46 x^{5} - 17 x^{4} + 19 x^{3} + 9 x^{2} - 2 x - 1 \)

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

Invariants

Degree:  $21$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[5, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(26730926988131422081122304\)\(\medspace = 2^{18}\cdot 317^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $16.25$
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:  $2, 317$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{1653148843589513} a^{20} + \frac{115197759085443}{1653148843589513} a^{19} - \frac{517727222659417}{1653148843589513} a^{18} + \frac{631439641891247}{1653148843589513} a^{17} - \frac{737187128448617}{1653148843589513} a^{16} - \frac{24010063922944}{1653148843589513} a^{15} - \frac{492799661287232}{1653148843589513} a^{14} - \frac{371685123185847}{1653148843589513} a^{13} - \frac{143095731596732}{1653148843589513} a^{12} + \frac{647207877155596}{1653148843589513} a^{11} - \frac{225915711041451}{1653148843589513} a^{10} - \frac{395040835575834}{1653148843589513} a^{9} - \frac{396273240181446}{1653148843589513} a^{8} - \frac{538888576067169}{1653148843589513} a^{7} + \frac{817593970203936}{1653148843589513} a^{6} + \frac{748350138872427}{1653148843589513} a^{5} - \frac{199976960190025}{1653148843589513} a^{4} - \frac{297442835221498}{1653148843589513} a^{3} - \frac{204628594825477}{1653148843589513} a^{2} + \frac{371059352421681}{1653148843589513} a + \frac{100790731293265}{1653148843589513}$

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:  $12$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
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:  \( 25357.2549184 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{5}\cdot(2\pi)^{8}\cdot 25357.2549184 \cdot 1}{2\sqrt{26730926988131422081122304}}\approx 0.190613719668$ (assuming GRH)

Galois group

$\PSL(2,7)$ (as 21T14):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 168
The 6 conjugacy class representatives for $\PSL(2,7)$
Character table for $\PSL(2,7)$

Intermediate fields

7.3.6431296.2, 7.3.6431296.1

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

Sibling fields

Degree 7 siblings: 7.3.6431296.2, 7.3.6431296.1
Degree 8 sibling: 8.0.646274503744.1
Degree 14 siblings: 14.2.4156382630830772224.1, 14.2.4156382630830772224.2
Degree 24 sibling: data not computed
Degree 28 sibling: 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 ${\href{/LocalNumberField/3.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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
2Data not computed
317Data not computed