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Properties

Label 28.0.55455050045...9861.1
Degree $28$
Signature $[0, 14]$
Discriminant $3^{14}\cdot 11^{14}\cdot 29^{27}$
Root discriminant $147.72$
Ramified primes $3, 11, 29$
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![2040675658868521, -1913132310046505, 1913132310046505, -1355130158950185, 1355130158950185, -629727362524969, 629727362524969, -189304236123945, 189304236123945, -37908786423593, 37908786423593, -5221132511017, 5221132511017, -506567042857, 506567042857, -35110496041, 35110496041, -1744545577, 1744545577, -61613865, 61613865, -1509161, 1509161, -24361, 24361, -233, 233, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 + 233*x^26 - 233*x^25 + 24361*x^24 - 24361*x^23 + 1509161*x^22 - 1509161*x^21 + 61613865*x^20 - 61613865*x^19 + 1744545577*x^18 - 1744545577*x^17 + 35110496041*x^16 - 35110496041*x^15 + 506567042857*x^14 - 506567042857*x^13 + 5221132511017*x^12 - 5221132511017*x^11 + 37908786423593*x^10 - 37908786423593*x^9 + 189304236123945*x^8 - 189304236123945*x^7 + 629727362524969*x^6 - 629727362524969*x^5 + 1355130158950185*x^4 - 1355130158950185*x^3 + 1913132310046505*x^2 - 1913132310046505*x + 2040675658868521)
 
gp: K = bnfinit(x^28 - x^27 + 233*x^26 - 233*x^25 + 24361*x^24 - 24361*x^23 + 1509161*x^22 - 1509161*x^21 + 61613865*x^20 - 61613865*x^19 + 1744545577*x^18 - 1744545577*x^17 + 35110496041*x^16 - 35110496041*x^15 + 506567042857*x^14 - 506567042857*x^13 + 5221132511017*x^12 - 5221132511017*x^11 + 37908786423593*x^10 - 37908786423593*x^9 + 189304236123945*x^8 - 189304236123945*x^7 + 629727362524969*x^6 - 629727362524969*x^5 + 1355130158950185*x^4 - 1355130158950185*x^3 + 1913132310046505*x^2 - 1913132310046505*x + 2040675658868521, 1)
 

Normalized defining polynomial

\( x^{28} - x^{27} + 233 x^{26} - 233 x^{25} + 24361 x^{24} - 24361 x^{23} + 1509161 x^{22} - 1509161 x^{21} + 61613865 x^{20} - 61613865 x^{19} + 1744545577 x^{18} - 1744545577 x^{17} + 35110496041 x^{16} - 35110496041 x^{15} + 506567042857 x^{14} - 506567042857 x^{13} + 5221132511017 x^{12} - 5221132511017 x^{11} + 37908786423593 x^{10} - 37908786423593 x^{9} + 189304236123945 x^{8} - 189304236123945 x^{7} + 629727362524969 x^{6} - 629727362524969 x^{5} + 1355130158950185 x^{4} - 1355130158950185 x^{3} + 1913132310046505 x^{2} - 1913132310046505 x + 2040675658868521 \)

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:  \(5545505004594863364258565003877268521046795476461201195989861=3^{14}\cdot 11^{14}\cdot 29^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $147.72$
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:  $3, 11, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(957=3\cdot 11\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{957}(1,·)$, $\chi_{957}(131,·)$, $\chi_{957}(98,·)$, $\chi_{957}(263,·)$, $\chi_{957}(265,·)$, $\chi_{957}(395,·)$, $\chi_{957}(397,·)$, $\chi_{957}(461,·)$, $\chi_{957}(529,·)$, $\chi_{957}(659,·)$, $\chi_{957}(661,·)$, $\chi_{957}(791,·)$, $\chi_{957}(100,·)$, $\chi_{957}(463,·)$, $\chi_{957}(199,·)$, $\chi_{957}(32,·)$, $\chi_{957}(34,·)$, $\chi_{957}(164,·)$, $\chi_{957}(230,·)$, $\chi_{957}(362,·)$, $\chi_{957}(67,·)$, $\chi_{957}(364,·)$, $\chi_{957}(760,·)$, $\chi_{957}(430,·)$, $\chi_{957}(626,·)$, $\chi_{957}(329,·)$, $\chi_{957}(824,·)$, $\chi_{957}(892,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{355262434568729} a^{15} + \frac{131983564182223}{355262434568729} a^{14} + \frac{120}{355262434568729} a^{13} - \frac{138863063477642}{355262434568729} a^{12} + \frac{5760}{355262434568729} a^{11} - \frac{70513405347855}{355262434568729} a^{10} + \frac{140800}{355262434568729} a^{9} - \frac{85127975262945}{355262434568729} a^{8} + \frac{1843200}{355262434568729} a^{7} + \frac{112353980761203}{355262434568729} a^{6} + \frac{12386304}{355262434568729} a^{5} - \frac{111303638411042}{355262434568729} a^{4} + \frac{36700160}{355262434568729} a^{3} + \frac{132655157746645}{355262434568729} a^{2} + \frac{31457280}{355262434568729} a + \frac{28815462343038}{355262434568729}$, $\frac{1}{355262434568729} a^{16} + \frac{128}{355262434568729} a^{14} + \frac{9918790248403}{355262434568729} a^{13} + \frac{6656}{355262434568729} a^{12} - \frac{34233117872275}{355262434568729} a^{11} + \frac{180224}{355262434568729} a^{10} + \frac{51725023383916}{355262434568729} a^{9} + \frac{2703360}{355262434568729} a^{8} - \frac{1614367821254}{355262434568729} a^{7} + \frac{22020096}{355262434568729} a^{6} + \frac{103353378524539}{355262434568729} a^{5} + \frac{88080384}{355262434568729} a^{4} + \frac{58151079529427}{355262434568729} a^{3} + \frac{134217728}{355262434568729} a^{2} + \frac{167961929338982}{355262434568729} a + \frac{33554432}{355262434568729}$, $\frac{1}{355262434568729} a^{17} + \frac{168619434222851}{355262434568729} a^{14} - \frac{8704}{355262434568729} a^{13} - \frac{22882721170549}{355262434568729} a^{12} - \frac{557056}{355262434568729} a^{11} - \frac{159382390877598}{355262434568729} a^{10} - \frac{15319040}{355262434568729} a^{9} - \frac{118369005794893}{355262434568729} a^{8} - \frac{213909504}{355262434568729} a^{7} - \frac{67458776160285}{355262434568729} a^{6} - \frac{1497366528}{355262434568729} a^{5} + \frac{94519413393643}{355262434568729} a^{4} - \frac{4563402752}{355262434568729} a^{3} - \frac{114563837501315}{355262434568729} a^{2} - \frac{3992977408}{355262434568729} a - \frac{135754834221574}{355262434568729}$, $\frac{1}{355262434568729} a^{18} - \frac{9792}{355262434568729} a^{14} - \frac{7256057495116}{355262434568729} a^{13} - \frac{678912}{355262434568729} a^{12} - \frac{119827403594272}{355262434568729} a^{11} - \frac{20680704}{355262434568729} a^{10} + \frac{98532210374648}{355262434568729} a^{9} - \frac{330891264}{355262434568729} a^{8} + \frac{155951944598520}{355262434568729} a^{7} - \frac{2807562240}{355262434568729} a^{6} + \frac{32684744790405}{355262434568729} a^{5} - \frac{11551113216}{355262434568729} a^{4} + \frac{70642632115921}{355262434568729} a^{3} - \frac{17968398336}{355262434568729} a^{2} + \frac{124497121041408}{355262434568729} a - \frac{4563402752}{355262434568729}$, $\frac{1}{355262434568729} a^{19} - \frac{68932546203602}{355262434568729} a^{14} + \frac{496128}{355262434568729} a^{13} + \frac{77654552429876}{355262434568729} a^{12} + \frac{35721216}{355262434568729} a^{11} - \frac{93822588781065}{355262434568729} a^{10} + \frac{1047822336}{355262434568729} a^{9} + \frac{28489668079314}{355262434568729} a^{8} + \frac{15241052160}{355262434568729} a^{7} - \frac{44895500863532}{355262434568729} a^{6} + \frac{109735575552}{355262434568729} a^{5} + \frac{130564568053229}{355262434568729} a^{4} + \frac{341399568384}{355262434568729} a^{3} - \frac{110921441652705}{355262434568729} a^{2} + \frac{303466283008}{355262434568729} a + \frac{82634215457270}{355262434568729}$, $\frac{1}{355262434568729} a^{20} + \frac{583680}{355262434568729} a^{14} - \frac{176738332787380}{355262434568729} a^{13} + \frac{45527040}{355262434568729} a^{12} + \frac{129504130696162}{355262434568729} a^{11} + \frac{1479278592}{355262434568729} a^{10} - \frac{38717282435366}{355262434568729} a^{9} + \frac{24654643200}{355262434568729} a^{8} + \frac{11905383534579}{355262434568729} a^{7} + \frac{215167795200}{355262434568729} a^{6} - \frac{13521891953184}{355262434568729} a^{5} + \frac{903704739840}{355262434568729} a^{4} - \frac{156266115653442}{355262434568729} a^{3} + \frac{1428076625920}{355262434568729} a^{2} - \frac{108487977072817}{355262434568729} a + \frac{367219703808}{355262434568729}$, $\frac{1}{355262434568729} a^{21} - \frac{171381025805473}{355262434568729} a^{14} - \frac{24514560}{355262434568729} a^{13} + \frac{18997643532288}{355262434568729} a^{12} - \frac{1882718208}{355262434568729} a^{11} + \frac{72671366316384}{355262434568729} a^{10} - \frac{57527500800}{355262434568729} a^{9} + \frac{149145642265510}{355262434568729} a^{8} - \frac{860671180800}{355262434568729} a^{7} + \frac{173571754472073}{355262434568729} a^{6} - \frac{6325933178880}{355262434568729} a^{5} + \frac{131041796143804}{355262434568729} a^{4} - \frac{19993072762880}{355262434568729} a^{3} + \frac{110865411952946}{355262434568729} a^{2} - \frac{17993765486592}{355262434568729} a - \frac{174883031651522}{355262434568729}$, $\frac{1}{355262434568729} a^{22} - \frac{29962240}{355262434568729} a^{14} - \frac{20500464797234}{355262434568729} a^{13} - \frac{2492858368}{355262434568729} a^{12} - \frac{46925660657027}{355262434568729} a^{11} - \frac{84373667840}{355262434568729} a^{10} + \frac{107235841084043}{355262434568729} a^{9} - \frac{1446405734400}{355262434568729} a^{8} - \frac{84396378165444}{355262434568729} a^{7} - \frac{12886160179200}{355262434568729} a^{6} + \frac{17492646453094}{355262434568729} a^{5} - \frac{54980950097920}{355262434568729} a^{4} + \frac{90212317671194}{355262434568729} a^{3} - \frac{87969520156672}{355262434568729} a^{2} + \frac{111924974308744}{355262434568729} a - \frac{22849226014720}{355262434568729}$, $\frac{1}{355262434568729} a^{23} + \frac{59753544700269}{355262434568729} a^{14} + \frac{1102610432}{355262434568729} a^{13} + \frac{59962538845084}{355262434568729} a^{12} + \frac{88208834560}{355262434568729} a^{11} - \frac{163357464785279}{355262434568729} a^{10} + \frac{2772277657600}{355262434568729} a^{9} - \frac{142568446501565}{355262434568729} a^{8} + \frac{42340240588800}{355262434568729} a^{7} + \frac{5437863312251}{355262434568729} a^{6} - \frac{39121971505689}{355262434568729} a^{5} + \frac{144340952442941}{355262434568729} a^{4} - \frac{54137821904459}{355262434568729} a^{3} - \frac{135080190716491}{355262434568729} a^{2} - \frac{146105956613707}{355262434568729} a - \frac{22652717629672}{355262434568729}$, $\frac{1}{355262434568729} a^{24} + \frac{1392771072}{355262434568729} a^{14} - \frac{5214133812616}{355262434568729} a^{13} + \frac{120706826240}{355262434568729} a^{12} - \frac{94475841236318}{355262434568729} a^{11} + \frac{4202189291520}{355262434568729} a^{10} - \frac{116686787736587}{355262434568729} a^{9} + \frac{73538312601600}{355262434568729} a^{8} + \frac{21286455718573}{355262434568729} a^{7} - \frac{44965999531058}{355262434568729} a^{6} - \frac{90208443772555}{355262434568729} a^{5} + \frac{33114840149816}{355262434568729} a^{4} - \frac{119516495550873}{355262434568729} a^{3} + \frac{28399367494843}{355262434568729} a^{2} - \frac{60064183854675}{355262434568729} a + \frac{151234629288373}{355262434568729}$, $\frac{1}{355262434568729} a^{25} - \frac{26070669063080}{355262434568729} a^{14} - \frac{46425702400}{355262434568729} a^{13} + \frac{56429252692817}{355262434568729} a^{12} - \frac{3820172083200}{355262434568729} a^{11} + \frac{128413522908231}{355262434568729} a^{10} - \frac{122563854336000}{355262434568729} a^{9} - \frac{56896676353259}{355262434568729} a^{8} - \frac{125284597460355}{355262434568729} a^{7} - \frac{42177193098378}{355262434568729} a^{6} - \frac{165574200749080}{355262434568729} a^{5} - \frac{100191448334415}{355262434568729} a^{4} + \frac{71268759620299}{355262434568729} a^{3} - \frac{21414089421759}{355262434568729} a^{2} + \frac{35724493437880}{355262434568729} a + \frac{106873737965717}{355262434568729}$, $\frac{1}{355262434568729} a^{26} - \frac{60353413120}{355262434568729} a^{14} - \frac{12452370856144}{355262434568729} a^{13} - \frac{5380075683840}{355262434568729} a^{12} + \frac{19457503676664}{355262434568729} a^{11} + \frac{164062821804569}{355262434568729} a^{10} + \frac{121833441202713}{355262434568729} a^{9} + \frac{153520118768890}{355262434568729} a^{8} - \frac{92384759464376}{355262434568729} a^{7} + \frac{114939144468632}{355262434568729} a^{6} + \frac{145044098378794}{355262434568729} a^{5} + \frac{146290195839847}{355262434568729} a^{4} - \frac{57946588233965}{355262434568729} a^{3} - \frac{169495068691753}{355262434568729} a^{2} + \frac{126311943922216}{355262434568729} a - \frac{154013514467324}{355262434568729}$, $\frac{1}{355262434568729} a^{27} - \frac{56035668852648}{355262434568729} a^{14} + \frac{1862333890560}{355262434568729} a^{13} + \frac{111802212030422}{355262434568729} a^{12} + \frac{156436046807040}{355262434568729} a^{11} - \frac{78148606056683}{355262434568729} a^{10} + \frac{124982256415394}{355262434568729} a^{9} + \frac{98189052529783}{355262434568729} a^{8} + \frac{161208187240455}{355262434568729} a^{7} + \frac{76843698387255}{355262434568729} a^{6} - \frac{125412229426218}{355262434568729} a^{5} + \frac{170422002759042}{355262434568729} a^{4} + \frac{104405879950861}{355262434568729} a^{3} - \frac{127475743229228}{355262434568729} a^{2} - \frac{122248378241500}{355262434568729} a + \frac{105637149068650}{355262434568729}$

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

\(\Q(\sqrt{29}) \), 4.0.26559621.1, 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 ${\href{/LocalNumberField/5.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/13.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{7}$ $28$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ 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.

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
3Data not computed
11Data not computed
29Data not computed