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Properties

Label 28.28.2311051677...0789.1
Degree $28$
Signature $[28, 0]$
Discriminant $29^{27}\cdot 31^{14}$
Root discriminant $143.17$
Ramified primes $29, 31$
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![-22496485878439, 150039834700455, -150039834700455, -407962316395865, 407962316395865, 317440480029351, -317440480029351, -122982646371673, 122982646371673, 28412803328679, -28412803328679, -4274850583897, 4274850583897, 439714884263, -439714884263, -31741662553, 31741662553, 1624287911, -1624287911, -58643801, 58643801, 1460903, -1460903, -23897, 23897, 231, -231, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 - 231*x^26 + 231*x^25 + 23897*x^24 - 23897*x^23 - 1460903*x^22 + 1460903*x^21 + 58643801*x^20 - 58643801*x^19 - 1624287911*x^18 + 1624287911*x^17 + 31741662553*x^16 - 31741662553*x^15 - 439714884263*x^14 + 439714884263*x^13 + 4274850583897*x^12 - 4274850583897*x^11 - 28412803328679*x^10 + 28412803328679*x^9 + 122982646371673*x^8 - 122982646371673*x^7 - 317440480029351*x^6 + 317440480029351*x^5 + 407962316395865*x^4 - 407962316395865*x^3 - 150039834700455*x^2 + 150039834700455*x - 22496485878439)
 
gp: K = bnfinit(x^28 - x^27 - 231*x^26 + 231*x^25 + 23897*x^24 - 23897*x^23 - 1460903*x^22 + 1460903*x^21 + 58643801*x^20 - 58643801*x^19 - 1624287911*x^18 + 1624287911*x^17 + 31741662553*x^16 - 31741662553*x^15 - 439714884263*x^14 + 439714884263*x^13 + 4274850583897*x^12 - 4274850583897*x^11 - 28412803328679*x^10 + 28412803328679*x^9 + 122982646371673*x^8 - 122982646371673*x^7 - 317440480029351*x^6 + 317440480029351*x^5 + 407962316395865*x^4 - 407962316395865*x^3 - 150039834700455*x^2 + 150039834700455*x - 22496485878439, 1)
 

Normalized defining polynomial

\( x^{28} - x^{27} - 231 x^{26} + 231 x^{25} + 23897 x^{24} - 23897 x^{23} - 1460903 x^{22} + 1460903 x^{21} + 58643801 x^{20} - 58643801 x^{19} - 1624287911 x^{18} + 1624287911 x^{17} + 31741662553 x^{16} - 31741662553 x^{15} - 439714884263 x^{14} + 439714884263 x^{13} + 4274850583897 x^{12} - 4274850583897 x^{11} - 28412803328679 x^{10} + 28412803328679 x^{9} + 122982646371673 x^{8} - 122982646371673 x^{7} - 317440480029351 x^{6} + 317440480029351 x^{5} + 407962316395865 x^{4} - 407962316395865 x^{3} - 150039834700455 x^{2} + 150039834700455 x - 22496485878439 \)

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

Invariants

Degree:  $28$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[28, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2311051677985488109296342199847259892756218483742420538210789=29^{27}\cdot 31^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $143.17$
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:  $29, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(899=29\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{899}(1,·)$, $\chi_{899}(619,·)$, $\chi_{899}(342,·)$, $\chi_{899}(373,·)$, $\chi_{899}(776,·)$, $\chi_{899}(650,·)$, $\chi_{899}(247,·)$, $\chi_{899}(588,·)$, $\chi_{899}(528,·)$, $\chi_{899}(497,·)$, $\chi_{899}(340,·)$, $\chi_{899}(278,·)$, $\chi_{899}(495,·)$, $\chi_{899}(94,·)$, $\chi_{899}(867,·)$, $\chi_{899}(869,·)$, $\chi_{899}(807,·)$, $\chi_{899}(681,·)$, $\chi_{899}(683,·)$, $\chi_{899}(125,·)$, $\chi_{899}(743,·)$, $\chi_{899}(433,·)$, $\chi_{899}(309,·)$, $\chi_{899}(745,·)$, $\chi_{899}(185,·)$, $\chi_{899}(187,·)$, $\chi_{899}(61,·)$, $\chi_{899}(63,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{1908200831081} a^{15} + \frac{130313736210}{1908200831081} a^{14} - \frac{120}{1908200831081} a^{13} + \frac{670468193128}{1908200831081} a^{12} + \frac{5760}{1908200831081} a^{11} - \frac{877588031417}{1908200831081} a^{10} - \frac{140800}{1908200831081} a^{9} + \frac{585785328583}{1908200831081} a^{8} + \frac{1843200}{1908200831081} a^{7} + \frac{308727311762}{1908200831081} a^{6} - \frac{12386304}{1908200831081} a^{5} + \frac{897722112044}{1908200831081} a^{4} + \frac{36700160}{1908200831081} a^{3} + \frac{112756606993}{1908200831081} a^{2} - \frac{31457280}{1908200831081} a + \frac{80010138395}{1908200831081}$, $\frac{1}{1908200831081} a^{16} - \frac{128}{1908200831081} a^{14} - \frac{865690941401}{1908200831081} a^{13} + \frac{6656}{1908200831081} a^{12} + \frac{346418844897}{1908200831081} a^{11} - \frac{180224}{1908200831081} a^{10} - \frac{499347978313}{1908200831081} a^{9} + \frac{2703360}{1908200831081} a^{8} + \frac{809757360637}{1908200831081} a^{7} - \frac{22020096}{1908200831081} a^{6} - \frac{560998985315}{1908200831081} a^{5} + \frac{88080384}{1908200831081} a^{4} + \frac{335795110179}{1908200831081} a^{3} - \frac{134217728}{1908200831081} a^{2} + \frac{434034515973}{1908200831081} a + \frac{33554432}{1908200831081}$, $\frac{1}{1908200831081} a^{17} + \frac{548860644831}{1908200831081} a^{14} - \frac{8704}{1908200831081} a^{13} + \frac{297310166636}{1908200831081} a^{12} + \frac{557056}{1908200831081} a^{11} - \frac{246766965910}{1908200831081} a^{10} - \frac{15319040}{1908200831081} a^{9} - \frac{537753823979}{1908200831081} a^{8} + \frac{213909504}{1908200831081} a^{7} + \frac{792080298601}{1908200831081} a^{6} - \frac{1497366528}{1908200831081} a^{5} + \frac{752175586951}{1908200831081} a^{4} + \frac{4563402752}{1908200831081} a^{3} - \frac{398726437571}{1908200831081} a^{2} - \frac{3992977408}{1908200831081} a + \frac{700293559155}{1908200831081}$, $\frac{1}{1908200831081} a^{18} - \frac{9792}{1908200831081} a^{14} - \frac{626441541479}{1908200831081} a^{13} + \frac{678912}{1908200831081} a^{12} + \frac{204695908747}{1908200831081} a^{11} - \frac{20680704}{1908200831081} a^{10} + \frac{723781262483}{1908200831081} a^{9} + \frac{330891264}{1908200831081} a^{8} + \frac{236937026685}{1908200831081} a^{7} - \frac{2807562240}{1908200831081} a^{6} + \frac{818992778551}{1908200831081} a^{5} + \frac{11551113216}{1908200831081} a^{4} - \frac{596703301571}{1908200831081} a^{3} - \frac{17968398336}{1908200831081} a^{2} - \frac{765612698343}{1908200831081} a + \frac{4563402752}{1908200831081}$, $\frac{1}{1908200831081} a^{19} + \frac{727508264733}{1908200831081} a^{14} - \frac{496128}{1908200831081} a^{13} - \frac{689816731598}{1908200831081} a^{12} + \frac{35721216}{1908200831081} a^{11} + \frac{10119984962}{1908200831081} a^{10} - \frac{1047822336}{1908200831081} a^{9} + \frac{195176281935}{1908200831081} a^{8} + \frac{15241052160}{1908200831081} a^{7} - \frac{621487711330}{1908200831081} a^{6} - \frac{109735575552}{1908200831081} a^{5} + \frac{725189874191}{1908200831081} a^{4} + \frac{341399568384}{1908200831081} a^{3} + \frac{407002612295}{1908200831081} a^{2} - \frac{303466283008}{1908200831081} a - \frac{811266410451}{1908200831081}$, $\frac{1}{1908200831081} a^{20} - \frac{583680}{1908200831081} a^{14} + \frac{742137637717}{1908200831081} a^{13} + \frac{45527040}{1908200831081} a^{12} - \frac{28459823242}{1908200831081} a^{11} - \frac{1479278592}{1908200831081} a^{10} - \frac{769962570826}{1908200831081} a^{9} + \frac{24654643200}{1908200831081} a^{8} + \frac{390379480957}{1908200831081} a^{7} - \frac{215167795200}{1908200831081} a^{6} + \frac{489653200941}{1908200831081} a^{5} + \frac{903704739840}{1908200831081} a^{4} + \frac{751518527280}{1908200831081} a^{3} + \frac{480124205161}{1908200831081} a^{2} - \frac{16165291249}{1908200831081} a + \frac{367219703808}{1908200831081}$, $\frac{1}{1908200831081} a^{21} - \frac{529639029224}{1908200831081} a^{14} - \frac{24514560}{1908200831081} a^{13} - \frac{704535456925}{1908200831081} a^{12} + \frac{1882718208}{1908200831081} a^{11} + \frac{354352845011}{1908200831081} a^{10} - \frac{57527500800}{1908200831081} a^{9} + \frac{146053712817}{1908200831081} a^{8} + \frac{860671180800}{1908200831081} a^{7} - \frac{590299858053}{1908200831081} a^{6} - \frac{601330685637}{1908200831081} a^{5} + \frac{786665682005}{1908200831081} a^{4} + \frac{911064452070}{1908200831081} a^{3} - \frac{86459600699}{1908200831081} a^{2} - \frac{819958006863}{1908200831081} a + \frac{918639348287}{1908200831081}$, $\frac{1}{1908200831081} a^{22} - \frac{29962240}{1908200831081} a^{14} + \frac{617609292949}{1908200831081} a^{13} + \frac{2492858368}{1908200831081} a^{12} - \frac{137967723268}{1908200831081} a^{11} - \frac{84373667840}{1908200831081} a^{10} - \frac{540782380903}{1908200831081} a^{9} - \frac{461795096681}{1908200831081} a^{8} + \frac{247787272390}{1908200831081} a^{7} + \frac{471245638367}{1908200831081} a^{6} - \frac{723566039437}{1908200831081} a^{5} - \frac{356874003429}{1908200831081} a^{4} + \frac{783921707828}{1908200831081} a^{3} - \frac{192281926946}{1908200831081} a^{2} - \frac{917802248211}{1908200831081} a - \frac{49183958252}{1908200831081}$, $\frac{1}{1908200831081} a^{23} + \frac{211901980065}{1908200831081} a^{14} - \frac{1102610432}{1908200831081} a^{13} - \frac{495568626204}{1908200831081} a^{12} + \frac{88208834560}{1908200831081} a^{11} - \frac{71949300367}{1908200831081} a^{10} - \frac{864076826519}{1908200831081} a^{9} + \frac{427070048410}{1908200831081} a^{8} + \frac{359822305018}{1908200831081} a^{7} - \frac{823332299780}{1908200831081} a^{6} + \frac{620874896406}{1908200831081} a^{5} + \frac{43874542270}{1908200831081} a^{4} + \frac{303041328798}{1908200831081} a^{3} + \frac{562292082905}{1908200831081} a^{2} + \frac{71453488562}{1908200831081} a + \frac{723932989095}{1908200831081}$, $\frac{1}{1908200831081} a^{24} - \frac{1392771072}{1908200831081} a^{14} + \frac{126058177543}{1908200831081} a^{13} + \frac{120706826240}{1908200831081} a^{12} + \frac{621177417073}{1908200831081} a^{11} - \frac{385787629358}{1908200831081} a^{10} - \frac{402331582106}{1908200831081} a^{9} - \frac{881519810559}{1908200831081} a^{8} - \frac{374079124376}{1908200831081} a^{7} + \frac{403220440869}{1908200831081} a^{6} - \frac{150969566445}{1908200831081} a^{5} - \frac{444335739419}{1908200831081} a^{4} + \frac{220844597466}{1908200831081} a^{3} - \frac{341993206085}{1908200831081} a^{2} - \frac{73798212575}{1908200831081} a - \frac{410197235118}{1908200831081}$, $\frac{1}{1908200831081} a^{25} + \frac{630290887715}{1908200831081} a^{14} - \frac{46425702400}{1908200831081} a^{13} - \frac{515062020842}{1908200831081} a^{12} + \frac{3770421038}{1908200831081} a^{11} - \frac{21017921952}{1908200831081} a^{10} - \frac{439001146816}{1908200831081} a^{9} + \frac{326935855556}{1908200831081} a^{8} - \frac{879458283757}{1908200831081} a^{7} + \frac{393746315073}{1908200831081} a^{6} + \frac{313477866014}{1908200831081} a^{5} - \frac{776105939549}{1908200831081} a^{4} - \frac{396469601312}{1908200831081} a^{3} + \frac{11902030374}{1908200831081} a^{2} - \frac{908703419518}{1908200831081} a - \frac{495298242044}{1908200831081}$, $\frac{1}{1908200831081} a^{26} - \frac{60353413120}{1908200831081} a^{14} + \frac{700012092799}{1908200831081} a^{13} - \frac{344526809403}{1908200831081} a^{12} + \frac{809650386791}{1908200831081} a^{11} - \frac{379529656060}{1908200831081} a^{10} + \frac{587875043489}{1908200831081} a^{9} + \frac{598546763139}{1908200831081} a^{8} - \frac{940511238507}{1908200831081} a^{7} - \frac{592931844357}{1908200831081} a^{6} - \frac{587619002464}{1908200831081} a^{5} - \frac{14753085189}{1908200831081} a^{4} + \frac{796230209031}{1908200831081} a^{3} + \frac{236064379317}{1908200831081} a^{2} - \frac{622379300584}{1908200831081} a - \frac{607458974854}{1908200831081}$, $\frac{1}{1908200831081} a^{27} + \frac{287753170974}{1908200831081} a^{14} + \frac{45866940521}{1908200831081} a^{13} - \frac{722371145834}{1908200831081} a^{12} - \frac{36421341602}{1908200831081} a^{11} - \frac{698407439927}{1908200831081} a^{10} + \frac{56280270832}{1908200831081} a^{9} - \frac{458341795056}{1908200831081} a^{8} + \frac{434281410586}{1908200831081} a^{7} + \frac{118325890344}{1908200831081} a^{6} - \frac{887711532190}{1908200831081} a^{5} + \frac{211525499190}{1908200831081} a^{4} - \frac{216378582772}{1908200831081} a^{3} + \frac{547030119098}{1908200831081} a^{2} + \frac{52949397091}{1908200831081} a + \frac{163352263678}{1908200831081}$

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:  $27$
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

\(\Q(\sqrt{29}) \), 4.4.23437829.1, 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 $28$ $28$ ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{4}$ $28$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{7}$ $28$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R R $28$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{7}$ $28$ $28$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{28}$

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
29Data not computed
31Data not computed