Properties

Label 28.2.244...000.1
Degree $28$
Signature $[2, 13]$
Discriminant $-2.443\times 10^{51}$
Root discriminant $68.44$
Ramified primes $2, 5, 71$
Class number not computed
Class group not computed
Galois group $D_{28}$ (as 28T10)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 + 10*x^26 - 280*x^24 - 2600*x^22 + 24800*x^20 + 288000*x^18 - 1128000*x^16 - 72400000*x^14 - 791360000*x^12 - 3704000000*x^10 - 17401600000*x^8 - 19584000000*x^6 - 96256000000*x^4 - 10240000000*x^2 - 90880000000)
 
gp: K = bnfinit(x^28 + 10*x^26 - 280*x^24 - 2600*x^22 + 24800*x^20 + 288000*x^18 - 1128000*x^16 - 72400000*x^14 - 791360000*x^12 - 3704000000*x^10 - 17401600000*x^8 - 19584000000*x^6 - 96256000000*x^4 - 10240000000*x^2 - 90880000000, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-90880000000, 0, -10240000000, 0, -96256000000, 0, -19584000000, 0, -17401600000, 0, -3704000000, 0, -791360000, 0, -72400000, 0, -1128000, 0, 288000, 0, 24800, 0, -2600, 0, -280, 0, 10, 0, 1]);
 

\( x^{28} + 10 x^{26} - 280 x^{24} - 2600 x^{22} + 24800 x^{20} + 288000 x^{18} - 1128000 x^{16} - 72400000 x^{14} - 791360000 x^{12} - 3704000000 x^{10} - 17401600000 x^{8} - 19584000000 x^{6} - 96256000000 x^{4} - 10240000000 x^{2} - 90880000000 \)

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:  \(-2443365527501925442977500495872000000000000000000000\)\(\medspace = -\,2^{42}\cdot 5^{21}\cdot 71^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $68.44$
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:  $2, 5, 71$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{20} a^{4}$, $\frac{1}{20} a^{5}$, $\frac{1}{40} a^{6}$, $\frac{1}{40} a^{7}$, $\frac{1}{400} a^{8}$, $\frac{1}{400} a^{9}$, $\frac{1}{800} a^{10}$, $\frac{1}{800} a^{11}$, $\frac{1}{8000} a^{12}$, $\frac{1}{8000} a^{13}$, $\frac{1}{16000} a^{14}$, $\frac{1}{16000} a^{15}$, $\frac{1}{1120000} a^{16} + \frac{3}{112000} a^{14} + \frac{1}{56000} a^{12} + \frac{1}{2800} a^{10} + \frac{3}{2800} a^{8} - \frac{1}{140} a^{6} - \frac{3}{140} a^{4} + \frac{1}{14} a^{2} + \frac{3}{7}$, $\frac{1}{1120000} a^{17} + \frac{3}{112000} a^{15} + \frac{1}{56000} a^{13} + \frac{1}{2800} a^{11} + \frac{3}{2800} a^{9} - \frac{1}{140} a^{7} - \frac{3}{140} a^{5} + \frac{1}{14} a^{3} + \frac{3}{7} a$, $\frac{1}{2240000} a^{18} - \frac{1}{56000} a^{14} + \frac{1}{28000} a^{12} + \frac{1}{5600} a^{10} + \frac{1}{2800} a^{8} - \frac{1}{280} a^{6} + \frac{1}{140} a^{4} + \frac{1}{7} a^{2} - \frac{3}{7}$, $\frac{1}{2240000} a^{19} - \frac{1}{56000} a^{15} + \frac{1}{28000} a^{13} + \frac{1}{5600} a^{11} + \frac{1}{2800} a^{9} - \frac{1}{280} a^{7} + \frac{1}{140} a^{5} + \frac{1}{7} a^{3} - \frac{3}{7} a$, $\frac{1}{22400000} a^{20} - \frac{1}{56000} a^{14} + \frac{3}{56000} a^{12} - \frac{1}{1400} a^{8} - \frac{1}{280} a^{6} + \frac{3}{140} a^{4} - \frac{1}{7}$, $\frac{1}{22400000} a^{21} - \frac{1}{56000} a^{15} + \frac{3}{56000} a^{13} - \frac{1}{1400} a^{9} - \frac{1}{280} a^{7} + \frac{3}{140} a^{5} - \frac{1}{7} a$, $\frac{1}{44800000} a^{22} - \frac{1}{56000} a^{14} + \frac{3}{56000} a^{12} - \frac{3}{5600} a^{10} - \frac{3}{2800} a^{8} - \frac{3}{280} a^{6} - \frac{1}{70} a^{4} + \frac{1}{7} a^{2} + \frac{2}{7}$, $\frac{1}{44800000} a^{23} - \frac{1}{56000} a^{15} + \frac{3}{56000} a^{13} - \frac{3}{5600} a^{11} - \frac{3}{2800} a^{9} - \frac{3}{280} a^{7} - \frac{1}{70} a^{5} + \frac{1}{7} a^{3} + \frac{2}{7} a$, $\frac{1}{40768000000} a^{24} - \frac{43}{4076800000} a^{22} - \frac{23}{2038400000} a^{20} + \frac{31}{203840000} a^{18} + \frac{1}{1274000} a^{14} + \frac{101}{5096000} a^{12} - \frac{101}{254800} a^{10} + \frac{33}{63700} a^{8} + \frac{19}{6370} a^{6} - \frac{17}{3185} a^{4} + \frac{271}{1274} a^{2} - \frac{178}{637}$, $\frac{1}{40768000000} a^{25} - \frac{43}{4076800000} a^{23} - \frac{23}{2038400000} a^{21} + \frac{31}{203840000} a^{19} + \frac{1}{1274000} a^{15} + \frac{101}{5096000} a^{13} - \frac{101}{254800} a^{11} + \frac{33}{63700} a^{9} + \frac{19}{6370} a^{7} - \frac{17}{3185} a^{5} + \frac{271}{1274} a^{3} - \frac{178}{637} a$, $\frac{1}{12013927979311236518304128000000} a^{26} - \frac{769079677129429317}{107267214100993183199144000000} a^{24} - \frac{217511656150610514179}{42906885640397273279657600000} a^{22} - \frac{6058747029483599664607}{300348199482780912957603200000} a^{20} + \frac{19826373471189467779}{150174099741390456478801600} a^{18} + \frac{381730111821791694919}{938588123383690352992510000} a^{16} + \frac{5605738803562426732347}{750870498706952282394008000} a^{14} - \frac{778541379741508756591}{15017409974139045647880160} a^{12} - \frac{6796658613818907916109}{18771762467673807059850200} a^{10} + \frac{7624096038230621431899}{37543524935347614119700400} a^{8} + \frac{2209279803260204442773}{750870498706952282394008} a^{6} + \frac{1372647680754925492917}{268168035252482957997860} a^{4} - \frac{33268151691021561836269}{187717624676738070598502} a^{2} - \frac{9170651213542823661245}{93858812338369035299251}$, $\frac{1}{12013927979311236518304128000000} a^{27} - \frac{769079677129429317}{107267214100993183199144000000} a^{25} - \frac{217511656150610514179}{42906885640397273279657600000} a^{23} - \frac{6058747029483599664607}{300348199482780912957603200000} a^{21} + \frac{19826373471189467779}{150174099741390456478801600} a^{19} + \frac{381730111821791694919}{938588123383690352992510000} a^{17} + \frac{5605738803562426732347}{750870498706952282394008000} a^{15} - \frac{778541379741508756591}{15017409974139045647880160} a^{13} - \frac{6796658613818907916109}{18771762467673807059850200} a^{11} + \frac{7624096038230621431899}{37543524935347614119700400} a^{9} + \frac{2209279803260204442773}{750870498706952282394008} a^{7} + \frac{1372647680754925492917}{268168035252482957997860} a^{5} - \frac{33268151691021561836269}{187717624676738070598502} a^{3} - \frac{9170651213542823661245}{93858812338369035299251} a$

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

\(\Q(\sqrt{5}) \), 4.2.568000.1, 7.1.357911.1, 14.2.10007834681328125.1

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 $28$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/29.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $28$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $28$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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

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
5Data not computed
$71$$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
1.71.2t1.a.a$1$ $ 71 $ \(\Q(\sqrt{-71}) \) $C_2$ (as 2T1) $1$ $-1$
1.355.2t1.a.a$1$ $ 5 \cdot 71 $ \(\Q(\sqrt{-355}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.113600.4t3.c.a$2$ $ 2^{6} \cdot 5^{2} \cdot 71 $ 4.2.568000.1 $D_{4}$ (as 4T3) $1$ $0$
* 2.1775.14t3.b.b$2$ $ 5^{2} \cdot 71 $ 14.2.10007834681328125.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.1775.14t3.b.a$2$ $ 5^{2} \cdot 71 $ 14.2.10007834681328125.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.71.7t2.a.a$2$ $ 71 $ 7.1.357911.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.71.7t2.a.c$2$ $ 71 $ 7.1.357911.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.1775.14t3.b.c$2$ $ 5^{2} \cdot 71 $ 14.2.10007834681328125.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.71.7t2.a.b$2$ $ 71 $ 7.1.357911.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.113600.28t10.a.b$2$ $ 2^{6} \cdot 5^{2} \cdot 71 $ 28.2.2443365527501925442977500495872000000000000000000000.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.113600.28t10.a.e$2$ $ 2^{6} \cdot 5^{2} \cdot 71 $ 28.2.2443365527501925442977500495872000000000000000000000.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.113600.28t10.a.f$2$ $ 2^{6} \cdot 5^{2} \cdot 71 $ 28.2.2443365527501925442977500495872000000000000000000000.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.113600.28t10.a.d$2$ $ 2^{6} \cdot 5^{2} \cdot 71 $ 28.2.2443365527501925442977500495872000000000000000000000.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.113600.28t10.a.a$2$ $ 2^{6} \cdot 5^{2} \cdot 71 $ 28.2.2443365527501925442977500495872000000000000000000000.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.113600.28t10.a.c$2$ $ 2^{6} \cdot 5^{2} \cdot 71 $ 28.2.2443365527501925442977500495872000000000000000000000.1 $D_{28}$ (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 *.