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)

Related objects

Downloads

Learn more about

Show commands for: SageMath / Pari/GP / Magma

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(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)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1565568, 0, 23099904, 0, -105026816, 0, 141558144, 0, -75391104, 0, 11464832, 0, 2661648, 0, -906528, 0, -27616, 0, 35040, 0, -1656, 0, -160, 0, -24, 0, 0, 0, 1]);
 

\( x^{28} - 24 x^{24} - 160 x^{22} - 1656 x^{20} + 35040 x^{18} - 27616 x^{16} - 906528 x^{14} + 2661648 x^{12} + 11464832 x^{10} - 75391104 x^{8} + 141558144 x^{6} - 105026816 x^{4} + 23099904 x^{2} - 1565568 \)

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

Invariants

Degree:  $28$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 13]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-3206343011700717904956462085952501156673661196828672\)\(\medspace = -\,2^{77}\cdot 151^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $69.10$
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:  $2, 151$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{6} a^{7} + \frac{1}{6} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{12} a^{8} - \frac{1}{6} a^{6} - \frac{1}{6} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{12} a^{9} - \frac{1}{3} a$, $\frac{1}{12} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{12} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{24} a^{12} - \frac{1}{6} a^{4}$, $\frac{1}{24} a^{13} - \frac{1}{6} a^{5}$, $\frac{1}{72} a^{14} - \frac{1}{72} a^{12} - \frac{1}{36} a^{8} - \frac{1}{6} a^{6} + \frac{1}{9} a^{4} + \frac{2}{9} a^{2}$, $\frac{1}{72} a^{15} - \frac{1}{72} a^{13} - \frac{1}{36} a^{9} - \frac{2}{9} a^{5} - \frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{144} a^{16} + \frac{1}{72} a^{12} + \frac{1}{36} a^{10} + \frac{1}{36} a^{8} + \frac{2}{9} a^{6} - \frac{1}{6} a^{4} + \frac{4}{9} a^{2}$, $\frac{1}{144} a^{17} + \frac{1}{72} a^{13} + \frac{1}{36} a^{11} + \frac{1}{36} a^{9} + \frac{1}{18} a^{7} + \frac{1}{6} a^{5} - \frac{2}{9} a^{3} + \frac{1}{3} a$, $\frac{1}{144} a^{18} + \frac{1}{36} a^{10} - \frac{2}{9} a^{2}$, $\frac{1}{144} a^{19} + \frac{1}{36} a^{11} - \frac{2}{9} a^{3}$, $\frac{1}{288} a^{20} + \frac{1}{72} a^{12} - \frac{1}{9} a^{4}$, $\frac{1}{864} a^{21} + \frac{1}{216} a^{15} - \frac{1}{72} a^{13} - \frac{1}{36} a^{11} + \frac{1}{54} a^{9} + \frac{1}{18} a^{7} + \frac{1}{6} a^{5} - \frac{1}{27} a^{3} - \frac{1}{3} a$, $\frac{1}{78624} a^{22} - \frac{5}{3744} a^{20} + \frac{3}{1456} a^{18} - \frac{103}{39312} a^{16} - \frac{5}{936} a^{14} - \frac{131}{6552} a^{12} + \frac{17}{756} a^{10} + \frac{67}{3276} a^{8} - \frac{17}{234} a^{6} + \frac{1021}{4914} a^{4} + \frac{44}{819} a^{2} + \frac{9}{91}$, $\frac{1}{78624} a^{23} - \frac{1}{5616} a^{21} + \frac{3}{1456} a^{19} - \frac{103}{39312} a^{17} - \frac{1}{1404} a^{15} + \frac{17}{2184} a^{13} - \frac{1}{189} a^{11} + \frac{383}{9828} a^{9} - \frac{2}{117} a^{7} + \frac{1021}{4914} a^{5} + \frac{41}{2457} a^{3} - \frac{64}{273} a$, $\frac{1}{1415232} a^{24} - \frac{1}{176904} a^{22} - \frac{5}{4536} a^{20} - \frac{1037}{353808} a^{18} - \frac{323}{353808} a^{16} - \frac{1}{273} a^{14} + \frac{1499}{176904} a^{12} + \frac{445}{88452} a^{10} + \frac{223}{14742} a^{8} + \frac{6401}{44226} a^{6} + \frac{1279}{22113} a^{4} - \frac{269}{819} a^{2} + \frac{100}{273}$, $\frac{1}{1415232} a^{25} - \frac{1}{176904} a^{23} + \frac{1}{18144} a^{21} - \frac{1037}{353808} a^{19} - \frac{323}{353808} a^{17} + \frac{19}{19656} a^{15} - \frac{479}{88452} a^{13} - \frac{503}{22113} a^{11} + \frac{248}{7371} a^{9} + \frac{1487}{44226} a^{7} + \frac{1279}{22113} a^{5} - \frac{79}{2457} a^{3} + \frac{100}{273} a$, $\frac{1}{101081515688727484577604090607935936} a^{26} - \frac{9937481208340290893208335125}{101081515688727484577604090607935936} a^{24} - \frac{82966062596966047332367048817}{25270378922181871144401022651983984} a^{22} - \frac{2703041600651213493328674421115}{3610054131740267306343003235997712} a^{20} + \frac{2892143457537796605422602121555}{3610054131740267306343003235997712} a^{18} - \frac{68418116557086139107127404848705}{25270378922181871144401022651983984} a^{16} + \frac{79510949957139134754136137127169}{12635189461090935572200511325991992} a^{14} + \frac{238883814099308563993639536724987}{12635189461090935572200511325991992} a^{12} - \frac{7943766112218449192069163952793}{6317594730545467786100255662995996} a^{10} - \frac{17586658429596916673624468797945}{3158797365272733893050127831497998} a^{8} + \frac{30467189702287412084156137616029}{242984412713287222542317525499846} a^{6} - \frac{268527667271240114703609135179071}{3158797365272733893050127831497998} a^{4} - \frac{137695289971381110203566924220}{6499583056116736405452937924893} a^{2} - \frac{676273727539078448743788132626}{1499903782180785324335293367283}$, $\frac{1}{101081515688727484577604090607935936} a^{27} - \frac{9937481208340290893208335125}{101081515688727484577604090607935936} a^{25} - \frac{82966062596966047332367048817}{25270378922181871144401022651983984} a^{23} + \frac{2950523585133376963210714204061}{7220108263480534612686006471995424} a^{21} + \frac{2892143457537796605422602121555}{3610054131740267306343003235997712} a^{19} - \frac{68418116557086139107127404848705}{25270378922181871144401022651983984} a^{17} - \frac{37481545052962120544016745520905}{12635189461090935572200511325991992} a^{15} + \frac{238883814099308563993639536724987}{12635189461090935572200511325991992} a^{13} - \frac{45858127156842583034824621981226}{1579398682636366946525063915748999} a^{11} + \frac{257307920666059304898133269024295}{6317594730545467786100255662995996} a^{9} + \frac{3468921623033276246120857004935}{242984412713287222542317525499846} a^{7} + \frac{433427302789367417085308160709373}{3158797365272733893050127831497998} a^{5} + \frac{7530848976450756720498445579987}{19498749168350209216358813774679} a^{3} - \frac{176305800145483340632023676865}{1499903782180785324335293367283} a$

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

Class group and class number

not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $14$
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:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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.

Sibling fields

Degree 28 sibling: Deg 28

Frobenius cycle types

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

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

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
$151$$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.8.2t1.a.a$1$ $ 2^{3}$ $x^{2} - 2$ $C_2$ (as 2T1) $1$ $1$
1.151.2t1.a.a$1$ $ 151 $ $x^{2} - x + 38$ $C_2$ (as 2T1) $1$ $-1$
1.1208.2t1.b.a$1$ $ 2^{3} \cdot 151 $ $x^{2} + 302$ $C_2$ (as 2T1) $1$ $-1$
* 2.38656.4t3.a.a$2$ $ 2^{8} \cdot 151 $ $x^{4} + 12 x^{2} - 302$ $D_{4}$ (as 4T3) $1$ $0$
* 2.151.7t2.a.c$2$ $ 151 $ $x^{7} - x^{6} + x^{5} + 3 x^{3} - x^{2} + 3 x + 1$ $D_{7}$ (as 7T2) $1$ $0$
* 2.9664.14t3.a.a$2$ $ 2^{6} \cdot 151 $ $x^{14} + 2 x^{12} + 28 x^{10} + 80 x^{8} + 272 x^{6} + 544 x^{4} + 704 x^{2} - 128$ $D_{14}$ (as 14T3) $1$ $0$
* 2.9664.14t3.a.b$2$ $ 2^{6} \cdot 151 $ $x^{14} + 2 x^{12} + 28 x^{10} + 80 x^{8} + 272 x^{6} + 544 x^{4} + 704 x^{2} - 128$ $D_{14}$ (as 14T3) $1$ $0$
* 2.151.7t2.a.a$2$ $ 151 $ $x^{7} - x^{6} + x^{5} + 3 x^{3} - x^{2} + 3 x + 1$ $D_{7}$ (as 7T2) $1$ $0$
* 2.151.7t2.a.b$2$ $ 151 $ $x^{7} - x^{6} + x^{5} + 3 x^{3} - x^{2} + 3 x + 1$ $D_{7}$ (as 7T2) $1$ $0$
* 2.9664.14t3.a.c$2$ $ 2^{6} \cdot 151 $ $x^{14} + 2 x^{12} + 28 x^{10} + 80 x^{8} + 272 x^{6} + 544 x^{4} + 704 x^{2} - 128$ $D_{14}$ (as 14T3) $1$ $0$
* 2.38656.28t10.a.b$2$ $ 2^{8} \cdot 151 $ $x^{28} - 24 x^{24} - 160 x^{22} - 1656 x^{20} + 35040 x^{18} - 27616 x^{16} - 906528 x^{14} + 2661648 x^{12} + 11464832 x^{10} - 75391104 x^{8} + 141558144 x^{6} - 105026816 x^{4} + 23099904 x^{2} - 1565568$ $D_{28}$ (as 28T10) $1$ $0$
* 2.38656.28t10.a.f$2$ $ 2^{8} \cdot 151 $ $x^{28} - 24 x^{24} - 160 x^{22} - 1656 x^{20} + 35040 x^{18} - 27616 x^{16} - 906528 x^{14} + 2661648 x^{12} + 11464832 x^{10} - 75391104 x^{8} + 141558144 x^{6} - 105026816 x^{4} + 23099904 x^{2} - 1565568$ $D_{28}$ (as 28T10) $1$ $0$
* 2.38656.28t10.a.c$2$ $ 2^{8} \cdot 151 $ $x^{28} - 24 x^{24} - 160 x^{22} - 1656 x^{20} + 35040 x^{18} - 27616 x^{16} - 906528 x^{14} + 2661648 x^{12} + 11464832 x^{10} - 75391104 x^{8} + 141558144 x^{6} - 105026816 x^{4} + 23099904 x^{2} - 1565568$ $D_{28}$ (as 28T10) $1$ $0$
* 2.38656.28t10.a.d$2$ $ 2^{8} \cdot 151 $ $x^{28} - 24 x^{24} - 160 x^{22} - 1656 x^{20} + 35040 x^{18} - 27616 x^{16} - 906528 x^{14} + 2661648 x^{12} + 11464832 x^{10} - 75391104 x^{8} + 141558144 x^{6} - 105026816 x^{4} + 23099904 x^{2} - 1565568$ $D_{28}$ (as 28T10) $1$ $0$
* 2.38656.28t10.a.a$2$ $ 2^{8} \cdot 151 $ $x^{28} - 24 x^{24} - 160 x^{22} - 1656 x^{20} + 35040 x^{18} - 27616 x^{16} - 906528 x^{14} + 2661648 x^{12} + 11464832 x^{10} - 75391104 x^{8} + 141558144 x^{6} - 105026816 x^{4} + 23099904 x^{2} - 1565568$ $D_{28}$ (as 28T10) $1$ $0$
* 2.38656.28t10.a.e$2$ $ 2^{8} \cdot 151 $ $x^{28} - 24 x^{24} - 160 x^{22} - 1656 x^{20} + 35040 x^{18} - 27616 x^{16} - 906528 x^{14} + 2661648 x^{12} + 11464832 x^{10} - 75391104 x^{8} + 141558144 x^{6} - 105026816 x^{4} + 23099904 x^{2} - 1565568$ $D_{28}$ (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 *.