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Properties

Label 28.0.15909539583...6112.1
Degree $28$
Signature $[0, 14]$
Discriminant $2^{77}\cdot 29^{26}$
Root discriminant $153.38$
Ramified primes $2, 29$
Class number Not computed
Class group Not computed
Galois group $C_{28}$ (as 28T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![107648, 0, 2798848, 0, 26535232, 0, 117551616, 0, 281553344, 0, 392753728, 0, 331475104, 0, 173227904, 0, 56876888, 0, 11760544, 0, 1511944, 0, 117160, 0, 5162, 0, 116, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 + 116*x^26 + 5162*x^24 + 117160*x^22 + 1511944*x^20 + 11760544*x^18 + 56876888*x^16 + 173227904*x^14 + 331475104*x^12 + 392753728*x^10 + 281553344*x^8 + 117551616*x^6 + 26535232*x^4 + 2798848*x^2 + 107648)
 
gp: K = bnfinit(x^28 + 116*x^26 + 5162*x^24 + 117160*x^22 + 1511944*x^20 + 11760544*x^18 + 56876888*x^16 + 173227904*x^14 + 331475104*x^12 + 392753728*x^10 + 281553344*x^8 + 117551616*x^6 + 26535232*x^4 + 2798848*x^2 + 107648, 1)
 

Normalized defining polynomial

\( x^{28} + 116 x^{26} + 5162 x^{24} + 117160 x^{22} + 1511944 x^{20} + 11760544 x^{18} + 56876888 x^{16} + 173227904 x^{14} + 331475104 x^{12} + 392753728 x^{10} + 281553344 x^{8} + 117551616 x^{6} + 26535232 x^{4} + 2798848 x^{2} + 107648 \)

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

Invariants

Degree:  $28$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 14]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(15909539583539461596949590009846202679084771957405055149146112=2^{77}\cdot 29^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $153.38$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(464=2^{4}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{464}(1,·)$, $\chi_{464}(323,·)$, $\chi_{464}(51,·)$, $\chi_{464}(257,·)$, $\chi_{464}(393,·)$, $\chi_{464}(267,·)$, $\chi_{464}(115,·)$, $\chi_{464}(401,·)$, $\chi_{464}(67,·)$, $\chi_{464}(25,·)$, $\chi_{464}(281,·)$, $\chi_{464}(283,·)$, $\chi_{464}(297,·)$, $\chi_{464}(161,·)$, $\chi_{464}(35,·)$, $\chi_{464}(65,·)$, $\chi_{464}(81,·)$, $\chi_{464}(169,·)$, $\chi_{464}(299,·)$, $\chi_{464}(419,·)$, $\chi_{464}(49,·)$, $\chi_{464}(91,·)$, $\chi_{464}(179,·)$, $\chi_{464}(233,·)$, $\chi_{464}(313,·)$, $\chi_{464}(347,·)$, $\chi_{464}(187,·)$, $\chi_{464}(411,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{8} a^{12}$, $\frac{1}{8} a^{13}$, $\frac{1}{3944} a^{14} + \frac{1}{136} a^{12} + \frac{3}{34} a^{10} + \frac{1}{17} a^{8} + \frac{7}{34} a^{6} - \frac{1}{34} a^{4} - \frac{6}{17} a^{2} - \frac{4}{17}$, $\frac{1}{3944} a^{15} + \frac{1}{136} a^{13} + \frac{3}{34} a^{11} + \frac{1}{17} a^{9} + \frac{7}{34} a^{7} - \frac{1}{34} a^{5} - \frac{6}{17} a^{3} - \frac{4}{17} a$, $\frac{1}{7888} a^{16} + \frac{7}{17}$, $\frac{1}{7888} a^{17} + \frac{7}{17} a$, $\frac{1}{7888} a^{18} + \frac{7}{17} a^{2}$, $\frac{1}{7888} a^{19} + \frac{7}{17} a^{3}$, $\frac{1}{646816} a^{20} - \frac{9}{161704} a^{18} - \frac{1}{40426} a^{16} + \frac{1}{40426} a^{14} + \frac{21}{5576} a^{12} + \frac{143}{2788} a^{10} + \frac{339}{2788} a^{8} - \frac{261}{1394} a^{6} + \frac{10}{697} a^{4} - \frac{167}{697} a^{2} - \frac{55}{697}$, $\frac{1}{646816} a^{21} - \frac{9}{161704} a^{19} - \frac{1}{40426} a^{17} + \frac{1}{40426} a^{15} + \frac{21}{5576} a^{13} + \frac{143}{2788} a^{11} + \frac{339}{2788} a^{9} - \frac{261}{1394} a^{7} + \frac{10}{697} a^{5} - \frac{167}{697} a^{3} - \frac{55}{697} a$, $\frac{1}{646816} a^{22} + \frac{7}{323408} a^{16} + \frac{15}{161704} a^{14} + \frac{38}{697} a^{12} - \frac{335}{2788} a^{10} - \frac{331}{2788} a^{8} + \frac{95}{1394} a^{6} - \frac{135}{697} a^{4} + \frac{165}{697} a^{2} + \frac{193}{697}$, $\frac{1}{646816} a^{23} + \frac{7}{323408} a^{17} + \frac{15}{161704} a^{15} + \frac{38}{697} a^{13} - \frac{335}{2788} a^{11} - \frac{331}{2788} a^{9} + \frac{95}{1394} a^{7} - \frac{135}{697} a^{5} + \frac{165}{697} a^{3} + \frac{193}{697} a$, $\frac{1}{1297512896} a^{24} - \frac{209}{648756448} a^{22} - \frac{3}{11185456} a^{20} - \frac{346}{20273639} a^{18} - \frac{15513}{324378224} a^{16} - \frac{1883}{162189112} a^{14} + \frac{70603}{2796364} a^{12} - \frac{212697}{2796364} a^{10} + \frac{83167}{1398182} a^{8} + \frac{155351}{699091} a^{6} - \frac{59686}{699091} a^{4} - \frac{226218}{699091} a^{2} + \frac{340652}{699091}$, $\frac{1}{1297512896} a^{25} - \frac{209}{648756448} a^{23} - \frac{3}{11185456} a^{21} - \frac{346}{20273639} a^{19} - \frac{15513}{324378224} a^{17} - \frac{1883}{162189112} a^{15} + \frac{70603}{2796364} a^{13} - \frac{212697}{2796364} a^{11} + \frac{83167}{1398182} a^{9} + \frac{155351}{699091} a^{7} - \frac{59686}{699091} a^{5} - \frac{226218}{699091} a^{3} + \frac{340652}{699091} a$, $\frac{1}{5738950142010944} a^{26} + \frac{1728965}{5738950142010944} a^{24} - \frac{86574517}{1434737535502736} a^{22} - \frac{274957611}{358684383875684} a^{20} - \frac{74076924143}{1434737535502736} a^{18} - \frac{66088236903}{1434737535502736} a^{16} + \frac{177733425}{17496799213448} a^{14} + \frac{382677974351}{24736854060392} a^{12} - \frac{42916795289}{3092106757549} a^{10} + \frac{1241386027069}{12368427030196} a^{8} - \frac{990662959559}{6184213515098} a^{6} + \frac{149424870190}{3092106757549} a^{4} + \frac{522738787187}{3092106757549} a^{2} - \frac{1447807015172}{3092106757549}$, $\frac{1}{5738950142010944} a^{27} + \frac{1728965}{5738950142010944} a^{25} - \frac{86574517}{1434737535502736} a^{23} - \frac{274957611}{358684383875684} a^{21} - \frac{74076924143}{1434737535502736} a^{19} - \frac{66088236903}{1434737535502736} a^{17} + \frac{177733425}{17496799213448} a^{15} + \frac{382677974351}{24736854060392} a^{13} - \frac{42916795289}{3092106757549} a^{11} + \frac{1241386027069}{12368427030196} a^{9} - \frac{990662959559}{6184213515098} a^{7} + \frac{149424870190}{3092106757549} a^{5} + \frac{522738787187}{3092106757549} a^{3} - \frac{1447807015172}{3092106757549} a$

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

Class group and class number

Not computed

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $13$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

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

Intermediate fields

\(\Q(\sqrt{2}) \), 4.0.1722368.6, 7.7.594823321.1, 14.14.742003380228915810271232.1

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 R $28$ $28$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{4}$ $28$ $28$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ $28$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{4}$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{4}$ $28$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{28}$ $28$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{4}$ $28$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{7}$

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

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
29Data not computed