Properties

Label 28.28.207...741.1
Degree $28$
Signature $[28, 0]$
Discriminant $2.071\times 10^{51}$
Root discriminant $68.03$
Ramified primes $7, 29$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{28}$ (as 28T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 - 57*x^26 + 57*x^25 + 1451*x^24 - 1451*x^23 - 21749*x^22 + 21749*x^21 + 213035*x^20 - 213035*x^19 - 1430453*x^18 + 1430453*x^17 + 6715531*x^16 - 6715531*x^15 - 22059893*x^14 + 22059893*x^13 + 49878667*x^12 - 49878667*x^11 - 74814837*x^10 + 74814837*x^9 + 69567115*x^8 - 69567115*x^7 - 35437941*x^6 + 35437941*x^5 + 7799435*x^4 - 7799435*x^3 - 515445*x^2 + 515445*x - 40309)
 
gp: K = bnfinit(x^28 - x^27 - 57*x^26 + 57*x^25 + 1451*x^24 - 1451*x^23 - 21749*x^22 + 21749*x^21 + 213035*x^20 - 213035*x^19 - 1430453*x^18 + 1430453*x^17 + 6715531*x^16 - 6715531*x^15 - 22059893*x^14 + 22059893*x^13 + 49878667*x^12 - 49878667*x^11 - 74814837*x^10 + 74814837*x^9 + 69567115*x^8 - 69567115*x^7 - 35437941*x^6 + 35437941*x^5 + 7799435*x^4 - 7799435*x^3 - 515445*x^2 + 515445*x - 40309, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-40309, 515445, -515445, -7799435, 7799435, 35437941, -35437941, -69567115, 69567115, 74814837, -74814837, -49878667, 49878667, 22059893, -22059893, -6715531, 6715531, 1430453, -1430453, -213035, 213035, 21749, -21749, -1451, 1451, 57, -57, -1, 1]);
 

\( x^{28} - x^{27} - 57 x^{26} + 57 x^{25} + 1451 x^{24} - 1451 x^{23} - 21749 x^{22} + 21749 x^{21} + 213035 x^{20} - 213035 x^{19} - 1430453 x^{18} + 1430453 x^{17} + 6715531 x^{16} - 6715531 x^{15} - 22059893 x^{14} + 22059893 x^{13} + 49878667 x^{12} - 49878667 x^{11} - 74814837 x^{10} + 74814837 x^{9} + 69567115 x^{8} - 69567115 x^{7} - 35437941 x^{6} + 35437941 x^{5} + 7799435 x^{4} - 7799435 x^{3} - 515445 x^{2} + 515445 x - 40309 \)

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:  \(2070706293589565601613551437543564286910572644210741\)\(\medspace = 7^{14}\cdot 29^{27}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $68.03$
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:  $7, 29$
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:  \(203=7\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{203}(64,·)$, $\chi_{203}(1,·)$, $\chi_{203}(195,·)$, $\chi_{203}(69,·)$, $\chi_{203}(71,·)$, $\chi_{203}(55,·)$, $\chi_{203}(76,·)$, $\chi_{203}(141,·)$, $\chi_{203}(78,·)$, $\chi_{203}(41,·)$, $\chi_{203}(22,·)$, $\chi_{203}(153,·)$, $\chi_{203}(90,·)$, $\chi_{203}(27,·)$, $\chi_{203}(92,·)$, $\chi_{203}(197,·)$, $\chi_{203}(160,·)$, $\chi_{203}(97,·)$, $\chi_{203}(36,·)$, $\chi_{203}(104,·)$, $\chi_{203}(169,·)$, $\chi_{203}(48,·)$, $\chi_{203}(118,·)$, $\chi_{203}(183,·)$, $\chi_{203}(120,·)$, $\chi_{203}(57,·)$, $\chi_{203}(188,·)$, $\chi_{203}(190,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{8641} a^{15} - \frac{4274}{8641} a^{14} - \frac{30}{8641} a^{13} - \frac{1302}{8641} a^{12} + \frac{360}{8641} a^{11} - \frac{2960}{8641} a^{10} - \frac{2200}{8641} a^{9} - \frac{351}{8641} a^{8} - \frac{1441}{8641} a^{7} + \frac{2711}{8641} a^{6} - \frac{3455}{8641} a^{5} + \frac{2146}{8641} a^{4} + \frac{319}{8641} a^{3} - \frac{1073}{8641} a^{2} - \frac{1920}{8641} a - \frac{3263}{8641}$, $\frac{1}{8641} a^{16} - \frac{32}{8641} a^{14} + \frac{93}{8641} a^{13} + \frac{416}{8641} a^{12} - \frac{2418}{8641} a^{11} - \frac{2816}{8641} a^{10} - \frac{1743}{8641} a^{9} + \frac{1919}{8641} a^{8} - \frac{3731}{8641} a^{7} - \frac{4222}{8641} a^{6} + \frac{2945}{8641} a^{5} + \frac{4222}{8641} a^{4} - \frac{2945}{8641} a^{3} + \frac{449}{8641} a^{2} - \frac{393}{8641} a + \frac{512}{8641}$, $\frac{1}{8641} a^{17} + \frac{1581}{8641} a^{14} - \frac{544}{8641} a^{13} - \frac{877}{8641} a^{12} + \frac{63}{8641} a^{11} - \frac{1412}{8641} a^{10} + \frac{647}{8641} a^{9} + \frac{2319}{8641} a^{8} + \frac{1512}{8641} a^{7} + \frac{3287}{8641} a^{6} - \frac{2646}{8641} a^{5} - \frac{3401}{8641} a^{4} + \frac{2016}{8641} a^{3} - \frac{165}{8641} a^{2} - \frac{441}{8641} a - \frac{724}{8641}$, $\frac{1}{8641} a^{18} - \frac{612}{8641} a^{14} + \frac{3348}{8641} a^{13} + \frac{1967}{8641} a^{12} - \frac{266}{8641} a^{11} - \frac{3015}{8641} a^{10} - \frac{1804}{8641} a^{9} + \frac{3419}{8641} a^{8} + \frac{284}{8641} a^{7} - \frac{2801}{8641} a^{6} - \frac{2158}{8641} a^{5} - \frac{3538}{8641} a^{4} - \frac{3326}{8641} a^{3} + \frac{2336}{8641} a^{2} + \frac{1805}{8641} a + \frac{126}{8641}$, $\frac{1}{8641} a^{19} - \frac{2758}{8641} a^{14} + \frac{889}{8641} a^{13} - \frac{2118}{8641} a^{12} + \frac{1280}{8641} a^{11} + \frac{1286}{8641} a^{10} - \frac{3626}{8641} a^{9} + \frac{1497}{8641} a^{8} - \frac{3311}{8641} a^{7} - \frac{2098}{8641} a^{6} - \frac{953}{8641} a^{5} - \frac{3406}{8641} a^{4} - \frac{1179}{8641} a^{3} + \frac{1845}{8641} a^{2} + \frac{262}{8641} a - \frac{885}{8641}$, $\frac{1}{8641} a^{20} - \frac{479}{8641} a^{14} + \frac{1552}{8641} a^{13} - \frac{3621}{8641} a^{12} + \frac{451}{8641} a^{11} - \frac{1561}{8641} a^{10} - \frac{121}{8641} a^{9} - \frac{3577}{8641} a^{8} - \frac{1516}{8641} a^{7} + \frac{1520}{8641} a^{6} - \frac{1273}{8641} a^{5} - \frac{1596}{8641} a^{4} + \frac{265}{8641} a^{3} - \frac{3850}{8641} a^{2} + \frac{688}{8641} a - \frac{4073}{8641}$, $\frac{1}{8641} a^{21} + \frac{2223}{8641} a^{14} - \frac{709}{8641} a^{13} - \frac{1055}{8641} a^{12} - \frac{1941}{8641} a^{11} - \frac{837}{8641} a^{10} - \frac{3175}{8641} a^{9} + \frac{3175}{8641} a^{8} + \frac{2561}{8641} a^{7} + \frac{1146}{8641} a^{6} + \frac{2531}{8641} a^{5} - \frac{80}{8641} a^{4} + \frac{2054}{8641} a^{3} - \frac{3460}{8641} a^{2} + \frac{834}{8641} a + \frac{1044}{8641}$, $\frac{1}{8641} a^{22} + \frac{3934}{8641} a^{14} - \frac{3493}{8641} a^{13} - \frac{2330}{8641} a^{12} + \frac{2496}{8641} a^{11} + \frac{1104}{8641} a^{10} + \frac{2969}{8641} a^{9} - \frac{3497}{8641} a^{8} - \frac{1322}{8641} a^{7} - \frac{1245}{8641} a^{6} - \frac{1464}{8641} a^{5} + \frac{1328}{8641} a^{4} - \frac{4035}{8641} a^{3} + \frac{1197}{8641} a^{2} + \frac{550}{8641} a + \frac{3850}{8641}$, $\frac{1}{8641} a^{23} + \frac{3678}{8641} a^{14} + \frac{3357}{8641} a^{13} + \frac{451}{8641} a^{12} + \frac{1988}{8641} a^{11} - \frac{459}{8641} a^{10} + \frac{1662}{8641} a^{9} - \frac{3048}{8641} a^{8} - \frac{847}{8641} a^{7} - \frac{3544}{8641} a^{6} + \frac{1005}{8641} a^{5} - \frac{4142}{8641} a^{4} - \frac{804}{8641} a^{3} - \frac{3717}{8641} a^{2} - \frac{3745}{8641} a - \frac{3884}{8641}$, $\frac{1}{8641} a^{24} - \frac{3491}{8641} a^{14} - \frac{1542}{8641} a^{13} + \frac{3630}{8641} a^{12} - \frac{2466}{8641} a^{11} + \frac{882}{8641} a^{10} + \frac{576}{8641} a^{9} + \frac{2622}{8641} a^{8} - \frac{479}{8641} a^{7} + \frac{1661}{8641} a^{6} + \frac{1078}{8641} a^{5} + \frac{4082}{8641} a^{4} - \frac{1823}{8641} a^{3} + \frac{2453}{8641} a^{2} - \frac{1821}{8641} a - \frac{1035}{8641}$, $\frac{1}{8641} a^{25} + \frac{931}{8641} a^{14} + \frac{2592}{8641} a^{13} - \frac{2582}{8641} a^{12} - \frac{3944}{8641} a^{11} + \frac{1852}{8641} a^{10} + \frac{4271}{8641} a^{9} + \frac{1202}{8641} a^{8} + \frac{192}{8641} a^{7} + \frac{3284}{8641} a^{6} - \frac{3128}{8641} a^{5} - \frac{1884}{8641} a^{4} + \frac{1393}{8641} a^{3} + \frac{2530}{8641} a^{2} + \frac{1661}{8641} a - \frac{2295}{8641}$, $\frac{1}{8641} a^{26} - \frac{1815}{8641} a^{14} - \frac{575}{8641} a^{13} - \frac{1522}{8641} a^{12} + \frac{3691}{8641} a^{11} + \frac{3552}{8641} a^{10} + \frac{1485}{8641} a^{9} - \frac{1385}{8641} a^{8} - \frac{3141}{8641} a^{7} - \frac{3897}{8641} a^{6} + \frac{269}{8641} a^{5} - \frac{462}{8641} a^{4} - \frac{665}{8641} a^{3} - \frac{1732}{8641} a^{2} - \frac{3462}{8641} a - \frac{3779}{8641}$, $\frac{1}{8641} a^{27} + \frac{1733}{8641} a^{14} - \frac{4126}{8641} a^{13} - \frac{446}{8641} a^{12} + \frac{236}{8641} a^{11} + \frac{3787}{8641} a^{10} - \frac{2243}{8641} a^{9} - \frac{772}{8641} a^{8} - \frac{1089}{8641} a^{7} + \frac{4005}{8641} a^{6} + \frac{2079}{8641} a^{5} - \frac{2766}{8641} a^{4} - \frac{1694}{8641} a^{3} + \frac{1909}{8641} a^{2} + \frac{2385}{8641} a - \frac{3260}{8641}$

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:  \( 40658806140034130 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{28}\cdot(2\pi)^{0}\cdot 40658806140034130 \cdot 1}{2\sqrt{2070706293589565601613551437543564286910572644210741}}\approx 0.119923764678369$ (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

\(\Q(\sqrt{29}) \), 4.4.1195061.1, 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 $28$ $28$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{4}$ R $28$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{7}$ $28$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{4}$ R $28$ $28$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{7}$ $28$ $28$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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
$7$7.14.7.1$x^{14} - 117649 x^{2} + 1647086$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
7.14.7.1$x^{14} - 117649 x^{2} + 1647086$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
29Data not computed