Properties

Label 28.2.149...000.1
Degree $28$
Signature $[2, 13]$
Discriminant $-1.491\times 10^{47}$
Root discriminant $48.39$
Ramified primes $2, 5, 71$
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 + 5*x^26 - 70*x^24 - 325*x^22 + 1550*x^20 + 9000*x^18 - 17625*x^16 - 565625*x^14 - 3091250*x^12 - 7234375*x^10 - 16993750*x^8 - 9562500*x^6 - 23500000*x^4 - 1250000*x^2 - 5546875)
 
gp: K = bnfinit(x^28 + 5*x^26 - 70*x^24 - 325*x^22 + 1550*x^20 + 9000*x^18 - 17625*x^16 - 565625*x^14 - 3091250*x^12 - 7234375*x^10 - 16993750*x^8 - 9562500*x^6 - 23500000*x^4 - 1250000*x^2 - 5546875, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5546875, 0, -1250000, 0, -23500000, 0, -9562500, 0, -16993750, 0, -7234375, 0, -3091250, 0, -565625, 0, -17625, 0, 9000, 0, 1550, 0, -325, 0, -70, 0, 5, 0, 1]);
 

\( x^{28} + 5 x^{26} - 70 x^{24} - 325 x^{22} + 1550 x^{20} + 9000 x^{18} - 17625 x^{16} - 565625 x^{14} - 3091250 x^{12} - 7234375 x^{10} - 16993750 x^{8} - 9562500 x^{6} - 23500000 x^{4} - 1250000 x^{2} - 5546875 \)

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:  \(-149131196746943691587982208000000000000000000000\)\(\medspace = -\,2^{28}\cdot 5^{21}\cdot 71^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $48.39$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{5} a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{25} a^{8}$, $\frac{1}{25} a^{9}$, $\frac{1}{25} a^{10}$, $\frac{1}{25} a^{11}$, $\frac{1}{125} a^{12}$, $\frac{1}{125} a^{13}$, $\frac{1}{125} a^{14}$, $\frac{1}{125} a^{15}$, $\frac{1}{4375} a^{16} + \frac{3}{875} a^{14} + \frac{1}{875} a^{12} + \frac{2}{175} a^{10} + \frac{3}{175} a^{8} - \frac{2}{35} a^{6} - \frac{3}{35} a^{4} + \frac{1}{7} a^{2} + \frac{3}{7}$, $\frac{1}{4375} a^{17} + \frac{3}{875} a^{15} + \frac{1}{875} a^{13} + \frac{2}{175} a^{11} + \frac{3}{175} a^{9} - \frac{2}{35} a^{7} - \frac{3}{35} a^{5} + \frac{1}{7} a^{3} + \frac{3}{7} a$, $\frac{1}{4375} a^{18} - \frac{2}{875} a^{14} + \frac{2}{875} a^{12} + \frac{1}{175} a^{10} + \frac{1}{175} a^{8} - \frac{1}{35} a^{6} + \frac{1}{35} a^{4} + \frac{2}{7} a^{2} - \frac{3}{7}$, $\frac{1}{4375} a^{19} - \frac{2}{875} a^{15} + \frac{2}{875} a^{13} + \frac{1}{175} a^{11} + \frac{1}{175} a^{9} - \frac{1}{35} a^{7} + \frac{1}{35} a^{5} + \frac{2}{7} a^{3} - \frac{3}{7} a$, $\frac{1}{21875} a^{20} - \frac{2}{875} a^{14} + \frac{3}{875} a^{12} - \frac{2}{175} a^{8} - \frac{1}{35} a^{6} + \frac{3}{35} a^{4} - \frac{1}{7}$, $\frac{1}{21875} a^{21} - \frac{2}{875} a^{15} + \frac{3}{875} a^{13} - \frac{2}{175} a^{9} - \frac{1}{35} a^{7} + \frac{3}{35} a^{5} - \frac{1}{7} a$, $\frac{1}{21875} a^{22} - \frac{2}{875} a^{14} + \frac{3}{875} a^{12} - \frac{3}{175} a^{10} - \frac{3}{175} a^{8} - \frac{3}{35} a^{6} - \frac{2}{35} a^{4} + \frac{2}{7} a^{2} + \frac{2}{7}$, $\frac{1}{21875} a^{23} - \frac{2}{875} a^{15} + \frac{3}{875} a^{13} - \frac{3}{175} a^{11} - \frac{3}{175} a^{9} - \frac{3}{35} a^{7} - \frac{2}{35} a^{5} + \frac{2}{7} a^{3} + \frac{2}{7} a$, $\frac{1}{9953125} a^{24} - \frac{43}{1990625} a^{22} - \frac{23}{1990625} a^{20} + \frac{31}{398125} a^{18} + \frac{8}{79625} a^{14} + \frac{101}{79625} a^{12} - \frac{202}{15925} a^{10} + \frac{132}{15925} a^{8} + \frac{76}{3185} a^{6} - \frac{68}{3185} a^{4} + \frac{271}{637} a^{2} - \frac{178}{637}$, $\frac{1}{9953125} a^{25} - \frac{43}{1990625} a^{23} - \frac{23}{1990625} a^{21} + \frac{31}{398125} a^{19} + \frac{8}{79625} a^{15} + \frac{101}{79625} a^{13} - \frac{202}{15925} a^{11} + \frac{132}{15925} a^{9} + \frac{76}{3185} a^{7} - \frac{68}{3185} a^{5} + \frac{271}{637} a^{3} - \frac{178}{637} a$, $\frac{1}{1466543942787016176550796875} a^{26} - \frac{6152637417035434536}{209506277541002310935828125} a^{24} - \frac{435023312301221028358}{41901255508200462187165625} a^{22} - \frac{6058747029483599664607}{293308788557403235310159375} a^{20} + \frac{158610987769515742232}{2346470308459225882481275} a^{18} + \frac{6107681789148667118704}{58661757711480647062031875} a^{16} + \frac{11211477607124853464694}{11732351542296129412406375} a^{14} - \frac{1557082759483017513182}{469294061691845176496255} a^{12} - \frac{27186634455275631664436}{2346470308459225882481275} a^{10} + \frac{7624096038230621431899}{2346470308459225882481275} a^{8} + \frac{2209279803260204442773}{93858812338369035299251} a^{6} + \frac{1372647680754925492917}{67042008813120739499465} a^{4} - \frac{33268151691021561836269}{93858812338369035299251} a^{2} - \frac{9170651213542823661245}{93858812338369035299251}$, $\frac{1}{1466543942787016176550796875} a^{27} - \frac{6152637417035434536}{209506277541002310935828125} a^{25} - \frac{435023312301221028358}{41901255508200462187165625} a^{23} - \frac{6058747029483599664607}{293308788557403235310159375} a^{21} + \frac{158610987769515742232}{2346470308459225882481275} a^{19} + \frac{6107681789148667118704}{58661757711480647062031875} a^{17} + \frac{11211477607124853464694}{11732351542296129412406375} a^{15} - \frac{1557082759483017513182}{469294061691845176496255} a^{13} - \frac{27186634455275631664436}{2346470308459225882481275} a^{11} + \frac{7624096038230621431899}{2346470308459225882481275} a^{9} + \frac{2209279803260204442773}{93858812338369035299251} a^{7} + \frac{1372647680754925492917}{67042008813120739499465} a^{5} - \frac{33268151691021561836269}{93858812338369035299251} a^{3} - \frac{9170651213542823661245}{93858812338369035299251} a$

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

\(\Q(\sqrt{5}) \), 4.2.142000.2, 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 ${\href{/LocalNumberField/7.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $28$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $28$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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) $1$ $-1$
* 2.28400.4t3.e.a$2$ $ 2^{4} \cdot 5^{2} \cdot 71 $ 4.2.142000.2 $D_{4}$ (as 4T3) $1$ $0$
* 2.1775.14t3.b.a$2$ $ 5^{2} \cdot 71 $ 14.2.10007834681328125.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.71.7t2.a.b$2$ $ 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.c$2$ $ 71 $ 7.1.357911.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.71.7t2.a.a$2$ $ 71 $ 7.1.357911.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.1775.14t3.b.b$2$ $ 5^{2} \cdot 71 $ 14.2.10007834681328125.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.28400.28t10.a.f$2$ $ 2^{4} \cdot 5^{2} \cdot 71 $ 28.2.149131196746943691587982208000000000000000000000.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.28400.28t10.a.e$2$ $ 2^{4} \cdot 5^{2} \cdot 71 $ 28.2.149131196746943691587982208000000000000000000000.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.28400.28t10.a.a$2$ $ 2^{4} \cdot 5^{2} \cdot 71 $ 28.2.149131196746943691587982208000000000000000000000.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.28400.28t10.a.c$2$ $ 2^{4} \cdot 5^{2} \cdot 71 $ 28.2.149131196746943691587982208000000000000000000000.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.28400.28t10.a.d$2$ $ 2^{4} \cdot 5^{2} \cdot 71 $ 28.2.149131196746943691587982208000000000000000000000.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.28400.28t10.a.b$2$ $ 2^{4} \cdot 5^{2} \cdot 71 $ 28.2.149131196746943691587982208000000000000000000000.1 $D_{28}$ (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 *.