Properties

Label 28.28.146...821.1
Degree $28$
Signature $[28, 0]$
Discriminant $1.460\times 10^{46}$
Root discriminant $44.54$
Ramified primes $3, 29$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{28}$ (as 28T1)

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 - x^27 - 28*x^26 + 28*x^25 + 349*x^24 - 349*x^23 - 2551*x^22 + 2551*x^21 + 12123*x^20 - 12123*x^19 - 39236*x^18 + 39236*x^17 + 88045*x^16 - 88045*x^15 - 136763*x^14 + 136763*x^13 + 144247*x^12 - 144247*x^11 - 99295*x^10 + 99295*x^9 + 41703*x^8 - 41703*x^7 - 9569*x^6 + 9569*x^5 + 987*x^4 - 987*x^3 - 28*x^2 + 28*x + 1)
 
gp: K = bnfinit(x^28 - x^27 - 28*x^26 + 28*x^25 + 349*x^24 - 349*x^23 - 2551*x^22 + 2551*x^21 + 12123*x^20 - 12123*x^19 - 39236*x^18 + 39236*x^17 + 88045*x^16 - 88045*x^15 - 136763*x^14 + 136763*x^13 + 144247*x^12 - 144247*x^11 - 99295*x^10 + 99295*x^9 + 41703*x^8 - 41703*x^7 - 9569*x^6 + 9569*x^5 + 987*x^4 - 987*x^3 - 28*x^2 + 28*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 28, -28, -987, 987, 9569, -9569, -41703, 41703, 99295, -99295, -144247, 144247, 136763, -136763, -88045, 88045, 39236, -39236, -12123, 12123, 2551, -2551, -349, 349, 28, -28, -1, 1]);
 

\( x^{28} - x^{27} - 28 x^{26} + 28 x^{25} + 349 x^{24} - 349 x^{23} - 2551 x^{22} + 2551 x^{21} + 12123 x^{20} - 12123 x^{19} - 39236 x^{18} + 39236 x^{17} + 88045 x^{16} - 88045 x^{15} - 136763 x^{14} + 136763 x^{13} + 144247 x^{12} - 144247 x^{11} - 99295 x^{10} + 99295 x^{9} + 41703 x^{8} - 41703 x^{7} - 9569 x^{6} + 9569 x^{5} + 987 x^{4} - 987 x^{3} - 28 x^{2} + 28 x + 1 \)

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

Invariants

Degree:  $28$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[28, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(14603047886206093768209337615200705673567789821\)\(\medspace = 3^{14}\cdot 29^{27}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $44.54$
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:  $3, 29$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $28$
This field is Galois and abelian over $\Q$.
Conductor:  \(87=3\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{87}(64,·)$, $\chi_{87}(1,·)$, $\chi_{87}(2,·)$, $\chi_{87}(67,·)$, $\chi_{87}(4,·)$, $\chi_{87}(68,·)$, $\chi_{87}(7,·)$, $\chi_{87}(8,·)$, $\chi_{87}(11,·)$, $\chi_{87}(13,·)$, $\chi_{87}(14,·)$, $\chi_{87}(77,·)$, $\chi_{87}(16,·)$, $\chi_{87}(17,·)$, $\chi_{87}(82,·)$, $\chi_{87}(22,·)$, $\chi_{87}(25,·)$, $\chi_{87}(26,·)$, $\chi_{87}(28,·)$, $\chi_{87}(32,·)$, $\chi_{87}(34,·)$, $\chi_{87}(41,·)$, $\chi_{87}(44,·)$, $\chi_{87}(47,·)$, $\chi_{87}(49,·)$, $\chi_{87}(50,·)$, $\chi_{87}(52,·)$, $\chi_{87}(56,·)$$\rbrace$
This is not a CM field.

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

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

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

Class group and class number

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

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $27$
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:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 114482618238342.14 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{28}\cdot(2\pi)^{0}\cdot 114482618238342.14 \cdot 1}{2\sqrt{14603047886206093768209337615200705673567789821}}\approx 0.127153313363501$ (assuming GRH)

Galois group

$C_{28}$ (as 28T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 28
The 28 conjugacy class representatives for $C_{28}$
Character table for $C_{28}$ is not computed

Intermediate fields

\(\Q(\sqrt{29}) \), 4.4.219501.1, 7.7.594823321.1, \(\Q(\zeta_{29})^+\)

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

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $28$ R ${\href{/LocalNumberField/5.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{4}$ $28$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{7}$ $28$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R $28$ $28$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{7}$ $28$ $28$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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
3Data not computed
29Data not computed