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Properties

Label 28.2.252...432.1
Degree $28$
Signature $[2, 13]$
Discriminant $-2.528\times 10^{53}$
Root discriminant \(80.77\)
Ramified primes $2,3,7$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_4\times F_7$ (as 28T41)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 3)
 
gp: K = bnfinit(y^28 - 3, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - 3);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 - 3)
 

\( x^{28} - 3 \) Copy content Toggle raw display

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

Invariants

Degree:  $28$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[2, 13]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-252754417691003413719324523325232135406322282693394432\) \(\medspace = -\,2^{56}\cdot 3^{27}\cdot 7^{28}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(80.77\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{2}3^{27/28}7^{47/42}\approx 101.82306150136618$
Ramified primes:   \(2\), \(3\), \(7\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-3}) \)
$\card{ \Aut(K/\Q) }$:  $2$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$ Copy content Toggle raw display

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

Monogenic:  Yes
Index:  $1$
Inessential primes:  None

Class group and class number

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

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $a^{14}-2$, $a^{16}+a^{4}+1$, $a^{21}+a^{14}+a^{7}+2$, $a^{26}+a^{24}+a^{22}+a^{20}+a^{18}+a^{16}+a^{14}+a^{12}+a^{10}+a^{8}+a^{6}+a^{4}+a^{2}+2$, $a^{27}+a^{26}+a^{25}+a^{24}+a^{23}+a^{22}+a^{21}+a^{20}+a^{19}+a^{18}+a^{17}+a^{16}+a^{15}+a^{14}+a^{13}+a^{12}+a^{11}+a^{10}+a^{9}+a^{8}+a^{7}+a^{6}+a^{5}+a^{4}+a^{3}+a^{2}+a+2$, $a^{26}+a^{22}-a^{20}-a^{18}+2a^{12}-a^{10}+a^{8}-a^{6}-2$, $a^{26}-2a^{24}-a^{20}+3a^{18}-2a^{16}-3a^{12}+3a^{10}-4a^{4}+a^{2}+1$, $a^{24}-a^{20}-a^{16}+2a^{12}+a^{8}-2a^{4}+1$, $2a^{26}-3a^{22}+4a^{20}-a^{18}-3a^{16}+6a^{14}-6a^{12}+2a^{10}+5a^{8}-8a^{6}+5a^{4}+a^{2}-5$, $5a^{27}+2a^{26}-12a^{25}-7a^{24}+4a^{23}+12a^{22}+7a^{21}-9a^{20}-9a^{19}-9a^{18}+13a^{17}+9a^{16}+7a^{15}-20a^{14}-9a^{13}-a^{12}+20a^{11}+10a^{10}-4a^{9}-19a^{8}-15a^{7}+15a^{6}+19a^{5}+13a^{4}-27a^{3}-14a^{2}-11a+29$, $a^{27}-7a^{26}+5a^{25}-15a^{24}+12a^{23}-15a^{22}+24a^{21}-12a^{20}+22a^{19}-19a^{18}+11a^{17}-16a^{16}+4a^{15}-6a^{14}-3a^{13}+14a^{12}-5a^{11}+27a^{10}-23a^{9}+25a^{8}-39a^{7}+24a^{6}-38a^{5}+31a^{4}-18a^{3}+32a^{2}-5a+8$, $70a^{27}+40a^{26}-2a^{25}-92a^{24}-56a^{23}-24a^{22}+34a^{21}+114a^{20}+46a^{19}+3a^{18}-91a^{17}-134a^{16}-3a^{15}+52a^{14}+142a^{13}+94a^{12}-66a^{11}-84a^{10}-152a^{9}-34a^{8}+106a^{7}+113a^{6}+166a^{5}-44a^{4}-171a^{3}-157a^{2}-113a+184$, $12a^{27}-42a^{26}+49a^{25}-31a^{24}+4a^{23}+21a^{22}-44a^{21}+46a^{20}-28a^{19}+4a^{18}+21a^{17}-50a^{16}+59a^{15}-42a^{14}+6a^{13}+38a^{12}-80a^{11}+87a^{10}-49a^{9}-22a^{8}+87a^{7}-118a^{6}+93a^{5}-18a^{4}-81a^{3}+137a^{2}-123a+55$, $7a^{27}+8a^{26}-17a^{25}+15a^{24}-4a^{23}-6a^{22}+12a^{21}-16a^{20}+21a^{19}-20a^{18}+6a^{17}+19a^{16}-36a^{15}+32a^{14}-7a^{13}-18a^{12}+29a^{11}-22a^{10}+8a^{9}+8a^{8}-22a^{7}+34a^{6}-37a^{5}+27a^{4}-2a^{3}-28a^{2}+52a-50$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 11582345197990588 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{13}\cdot 11582345197990588 \cdot 1}{2\cdot\sqrt{252754417691003413719324523325232135406322282693394432}}\cr\approx \mathstrut & 1.09601225756764 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^28 - 3)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^28 - 3, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^28 - 3);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 - 3);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$D_4\times F_7$ (as 28T41):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 336
The 35 conjugacy class representatives for $D_4\times F_7$
Character table for $D_4\times F_7$

Intermediate fields

\(\Q(\sqrt{3}) \), 4.2.6912.1, 7.1.600362847.1, 14.2.17716128058144132743168.1

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Degree 28 siblings: data not computed
Minimal sibling: 28.0.15426905376648157575642365925612312952046037762048.1

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 ${\href{/padicField/5.12.0.1}{12} }^{2}{,}\,{\href{/padicField/5.4.0.1}{4} }$ R ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{12}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ ${\href{/padicField/17.12.0.1}{12} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }$ ${\href{/padicField/19.6.0.1}{6} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ $28$ ${\href{/padicField/31.6.0.1}{6} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{7}$ ${\href{/padicField/43.14.0.1}{14} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.12.0.1}{12} }^{2}{,}\,{\href{/padicField/53.4.0.1}{4} }$ ${\href{/padicField/59.6.0.1}{6} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.4.8.5$x^{4} + 2 x^{2} + 4 x + 6$$4$$1$$8$$D_{4}$$[2, 3]^{2}$
2.12.24.313$x^{12} + 12 x^{11} + 66 x^{10} + 220 x^{9} + 510 x^{8} + 912 x^{7} + 1344 x^{6} + 1632 x^{5} + 1684 x^{4} + 1616 x^{3} + 1320 x^{2} + 688 x + 88$$4$$3$$24$$D_4 \times C_3$$[2, 3]^{6}$
2.12.24.313$x^{12} + 12 x^{11} + 66 x^{10} + 220 x^{9} + 510 x^{8} + 912 x^{7} + 1344 x^{6} + 1632 x^{5} + 1684 x^{4} + 1616 x^{3} + 1320 x^{2} + 688 x + 88$$4$$3$$24$$D_4 \times C_3$$[2, 3]^{6}$
\(3\) Copy content Toggle raw display Deg $28$$28$$1$$27$
\(7\) Copy content Toggle raw display 7.14.14.21$x^{14} - 14 x^{9} + 14 x^{8} + 14 x^{7} - 1127 x^{4} - 98 x^{3} - 49 x^{2} + 98 x + 49$$7$$2$$14$$F_7 \times C_2$$[7/6]_{6}^{2}$
7.14.14.21$x^{14} - 14 x^{9} + 14 x^{8} + 14 x^{7} - 1127 x^{4} - 98 x^{3} - 49 x^{2} + 98 x + 49$$7$$2$$14$$F_7 \times C_2$$[7/6]_{6}^{2}$