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Properties

Label 28.2.320...672.1
Degree $28$
Signature $[2, 13]$
Discriminant $-3.206\times 10^{51}$
Root discriminant \(69.10\)
Ramified primes $2,151$
Class number not computed
Class group not computed
Galois group $D_{28}$ (as 28T10)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 24*x^24 - 160*x^22 - 1656*x^20 + 35040*x^18 - 27616*x^16 - 906528*x^14 + 2661648*x^12 + 11464832*x^10 - 75391104*x^8 + 141558144*x^6 - 105026816*x^4 + 23099904*x^2 - 1565568)
 
gp: K = bnfinit(y^28 - 24*y^24 - 160*y^22 - 1656*y^20 + 35040*y^18 - 27616*y^16 - 906528*y^14 + 2661648*y^12 + 11464832*y^10 - 75391104*y^8 + 141558144*y^6 - 105026816*y^4 + 23099904*y^2 - 1565568, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - 24*x^24 - 160*x^22 - 1656*x^20 + 35040*x^18 - 27616*x^16 - 906528*x^14 + 2661648*x^12 + 11464832*x^10 - 75391104*x^8 + 141558144*x^6 - 105026816*x^4 + 23099904*x^2 - 1565568);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 - 24*x^24 - 160*x^22 - 1656*x^20 + 35040*x^18 - 27616*x^16 - 906528*x^14 + 2661648*x^12 + 11464832*x^10 - 75391104*x^8 + 141558144*x^6 - 105026816*x^4 + 23099904*x^2 - 1565568)
 

\( x^{28} - 24 x^{24} - 160 x^{22} - 1656 x^{20} + 35040 x^{18} - 27616 x^{16} - 906528 x^{14} + \cdots - 1565568 \) 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:   \(-3206343011700717904956462085952501156673661196828672\) \(\medspace = -\,2^{77}\cdot 151^{13}\) 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:  \(69.10\)
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:  $2^{11/4}151^{1/2}\approx 82.664865168866$
Ramified primes:   \(2\), \(151\) 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:  \(\Q(\sqrt{-302}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{6}a^{7}+\frac{1}{6}a^{5}-\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{12}a^{8}-\frac{1}{6}a^{6}-\frac{1}{6}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{12}a^{9}-\frac{1}{3}a$, $\frac{1}{12}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{12}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{24}a^{12}-\frac{1}{6}a^{4}$, $\frac{1}{24}a^{13}-\frac{1}{6}a^{5}$, $\frac{1}{72}a^{14}-\frac{1}{72}a^{12}-\frac{1}{36}a^{8}-\frac{1}{6}a^{6}+\frac{1}{9}a^{4}+\frac{2}{9}a^{2}$, $\frac{1}{72}a^{15}-\frac{1}{72}a^{13}-\frac{1}{36}a^{9}-\frac{2}{9}a^{5}-\frac{1}{9}a^{3}-\frac{1}{3}a$, $\frac{1}{144}a^{16}+\frac{1}{72}a^{12}+\frac{1}{36}a^{10}+\frac{1}{36}a^{8}+\frac{2}{9}a^{6}-\frac{1}{6}a^{4}+\frac{4}{9}a^{2}$, $\frac{1}{144}a^{17}+\frac{1}{72}a^{13}+\frac{1}{36}a^{11}+\frac{1}{36}a^{9}+\frac{1}{18}a^{7}+\frac{1}{6}a^{5}-\frac{2}{9}a^{3}+\frac{1}{3}a$, $\frac{1}{144}a^{18}+\frac{1}{36}a^{10}-\frac{2}{9}a^{2}$, $\frac{1}{144}a^{19}+\frac{1}{36}a^{11}-\frac{2}{9}a^{3}$, $\frac{1}{288}a^{20}+\frac{1}{72}a^{12}-\frac{1}{9}a^{4}$, $\frac{1}{864}a^{21}+\frac{1}{216}a^{15}-\frac{1}{72}a^{13}-\frac{1}{36}a^{11}+\frac{1}{54}a^{9}+\frac{1}{18}a^{7}+\frac{1}{6}a^{5}-\frac{1}{27}a^{3}-\frac{1}{3}a$, $\frac{1}{78624}a^{22}-\frac{5}{3744}a^{20}+\frac{3}{1456}a^{18}-\frac{103}{39312}a^{16}-\frac{5}{936}a^{14}-\frac{131}{6552}a^{12}+\frac{17}{756}a^{10}+\frac{67}{3276}a^{8}-\frac{17}{234}a^{6}+\frac{1021}{4914}a^{4}+\frac{44}{819}a^{2}+\frac{9}{91}$, $\frac{1}{78624}a^{23}-\frac{1}{5616}a^{21}+\frac{3}{1456}a^{19}-\frac{103}{39312}a^{17}-\frac{1}{1404}a^{15}+\frac{17}{2184}a^{13}-\frac{1}{189}a^{11}+\frac{383}{9828}a^{9}-\frac{2}{117}a^{7}+\frac{1021}{4914}a^{5}+\frac{41}{2457}a^{3}-\frac{64}{273}a$, $\frac{1}{1415232}a^{24}-\frac{1}{176904}a^{22}-\frac{5}{4536}a^{20}-\frac{1037}{353808}a^{18}-\frac{323}{353808}a^{16}-\frac{1}{273}a^{14}+\frac{1499}{176904}a^{12}+\frac{445}{88452}a^{10}+\frac{223}{14742}a^{8}+\frac{6401}{44226}a^{6}+\frac{1279}{22113}a^{4}-\frac{269}{819}a^{2}+\frac{100}{273}$, $\frac{1}{1415232}a^{25}-\frac{1}{176904}a^{23}+\frac{1}{18144}a^{21}-\frac{1037}{353808}a^{19}-\frac{323}{353808}a^{17}+\frac{19}{19656}a^{15}-\frac{479}{88452}a^{13}-\frac{503}{22113}a^{11}+\frac{248}{7371}a^{9}+\frac{1487}{44226}a^{7}+\frac{1279}{22113}a^{5}-\frac{79}{2457}a^{3}+\frac{100}{273}a$, $\frac{1}{10\!\cdots\!36}a^{26}-\frac{99\!\cdots\!25}{10\!\cdots\!36}a^{24}-\frac{82\!\cdots\!17}{25\!\cdots\!84}a^{22}-\frac{27\!\cdots\!15}{36\!\cdots\!12}a^{20}+\frac{28\!\cdots\!55}{36\!\cdots\!12}a^{18}-\frac{68\!\cdots\!05}{25\!\cdots\!84}a^{16}+\frac{79\!\cdots\!69}{12\!\cdots\!92}a^{14}+\frac{23\!\cdots\!87}{12\!\cdots\!92}a^{12}-\frac{79\!\cdots\!93}{63\!\cdots\!96}a^{10}-\frac{17\!\cdots\!45}{31\!\cdots\!98}a^{8}+\frac{30\!\cdots\!29}{24\!\cdots\!46}a^{6}-\frac{26\!\cdots\!71}{31\!\cdots\!98}a^{4}-\frac{13\!\cdots\!20}{64\!\cdots\!93}a^{2}-\frac{67\!\cdots\!26}{14\!\cdots\!83}$, $\frac{1}{10\!\cdots\!36}a^{27}-\frac{99\!\cdots\!25}{10\!\cdots\!36}a^{25}-\frac{82\!\cdots\!17}{25\!\cdots\!84}a^{23}+\frac{29\!\cdots\!61}{72\!\cdots\!24}a^{21}+\frac{28\!\cdots\!55}{36\!\cdots\!12}a^{19}-\frac{68\!\cdots\!05}{25\!\cdots\!84}a^{17}-\frac{37\!\cdots\!05}{12\!\cdots\!92}a^{15}+\frac{23\!\cdots\!87}{12\!\cdots\!92}a^{13}-\frac{45\!\cdots\!26}{15\!\cdots\!99}a^{11}+\frac{25\!\cdots\!95}{63\!\cdots\!96}a^{9}+\frac{34\!\cdots\!35}{24\!\cdots\!46}a^{7}+\frac{43\!\cdots\!73}{31\!\cdots\!98}a^{5}+\frac{75\!\cdots\!87}{19\!\cdots\!79}a^{3}-\frac{17\!\cdots\!65}{14\!\cdots\!83}a$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $3$

Class group and class number

not computed

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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^28 - 24*x^24 - 160*x^22 - 1656*x^20 + 35040*x^18 - 27616*x^16 - 906528*x^14 + 2661648*x^12 + 11464832*x^10 - 75391104*x^8 + 141558144*x^6 - 105026816*x^4 + 23099904*x^2 - 1565568)
 
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 - 24*x^24 - 160*x^22 - 1656*x^20 + 35040*x^18 - 27616*x^16 - 906528*x^14 + 2661648*x^12 + 11464832*x^10 - 75391104*x^8 + 141558144*x^6 - 105026816*x^4 + 23099904*x^2 - 1565568, 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 - 24*x^24 - 160*x^22 - 1656*x^20 + 35040*x^18 - 27616*x^16 - 906528*x^14 + 2661648*x^12 + 11464832*x^10 - 75391104*x^8 + 141558144*x^6 - 105026816*x^4 + 23099904*x^2 - 1565568);
 
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 - 24*x^24 - 160*x^22 - 1656*x^20 + 35040*x^18 - 27616*x^16 - 906528*x^14 + 2661648*x^12 + 11464832*x^10 - 75391104*x^8 + 141558144*x^6 - 105026816*x^4 + 23099904*x^2 - 1565568);
 
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

$D_{28}$ (as 28T10):

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 56
The 17 conjugacy class representatives for $D_{28}$
Character table for $D_{28}$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.2.309248.4, 7.1.3442951.1, 14.2.24859454395438333952.1

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 sibling: deg 28
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.2.0.1}{2} }^{14}$ $28$ ${\href{/padicField/7.2.0.1}{2} }^{13}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ $28$ ${\href{/padicField/13.2.0.1}{2} }^{14}$ ${\href{/padicField/17.7.0.1}{7} }^{4}$ $28$ ${\href{/padicField/23.2.0.1}{2} }^{13}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ $28$ ${\href{/padicField/31.7.0.1}{7} }^{4}$ $28$ ${\href{/padicField/41.2.0.1}{2} }^{13}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ $28$ ${\href{/padicField/47.14.0.1}{14} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{14}$ $28$

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 Deg $28$$4$$7$$77$
\(151\) Copy content Toggle raw display $\Q_{151}$$x + 145$$1$$1$$0$Trivial$[\ ]$
$\Q_{151}$$x + 145$$1$$1$$0$Trivial$[\ ]$
151.2.1.2$x^{2} + 151$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 151$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 151$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 151$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 151$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 151$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 151$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 151$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 151$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 151$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 151$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 151$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 151$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.8.2t1.a.a$1$ $ 2^{3}$ \(\Q(\sqrt{2}) \) $C_2$ (as 2T1) $1$ $1$
1.151.2t1.a.a$1$ $ 151 $ \(\Q(\sqrt{-151}) \) $C_2$ (as 2T1) $1$ $-1$
1.1208.2t1.b.a$1$ $ 2^{3} \cdot 151 $ \(\Q(\sqrt{-302}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.38656.4t3.a.a$2$ $ 2^{8} \cdot 151 $ 4.2.309248.4 $D_{4}$ (as 4T3) $1$ $0$
* 2.151.7t2.a.c$2$ $ 151 $ 7.1.3442951.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.9664.14t3.a.a$2$ $ 2^{6} \cdot 151 $ 14.2.24859454395438333952.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.9664.14t3.a.b$2$ $ 2^{6} \cdot 151 $ 14.2.24859454395438333952.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.151.7t2.a.a$2$ $ 151 $ 7.1.3442951.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.151.7t2.a.b$2$ $ 151 $ 7.1.3442951.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.9664.14t3.a.c$2$ $ 2^{6} \cdot 151 $ 14.2.24859454395438333952.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.38656.28t10.a.b$2$ $ 2^{8} \cdot 151 $ 28.2.3206343011700717904956462085952501156673661196828672.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.38656.28t10.a.f$2$ $ 2^{8} \cdot 151 $ 28.2.3206343011700717904956462085952501156673661196828672.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.38656.28t10.a.c$2$ $ 2^{8} \cdot 151 $ 28.2.3206343011700717904956462085952501156673661196828672.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.38656.28t10.a.d$2$ $ 2^{8} \cdot 151 $ 28.2.3206343011700717904956462085952501156673661196828672.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.38656.28t10.a.a$2$ $ 2^{8} \cdot 151 $ 28.2.3206343011700717904956462085952501156673661196828672.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.38656.28t10.a.e$2$ $ 2^{8} \cdot 151 $ 28.2.3206343011700717904956462085952501156673661196828672.1 $D_{28}$ (as 28T10) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.