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Properties

Label 28.0.554...861.1
Degree $28$
Signature $[0, 14]$
Discriminant $5.546\times 10^{60}$
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: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

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(y^28 - y^27 + 233*y^26 - 233*y^25 + 24361*y^24 - 24361*y^23 + 1509161*y^22 - 1509161*y^21 + 61613865*y^20 - 61613865*y^19 + 1744545577*y^18 - 1744545577*y^17 + 35110496041*y^16 - 35110496041*y^15 + 506567042857*y^14 - 506567042857*y^13 + 5221132511017*y^12 - 5221132511017*y^11 + 37908786423593*y^10 - 37908786423593*y^9 + 189304236123945*y^8 - 189304236123945*y^7 + 629727362524969*y^6 - 629727362524969*y^5 + 1355130158950185*y^4 - 1355130158950185*y^3 + 1913132310046505*y^2 - 1913132310046505*y + 2040675658868521, 1)
 
magma: R<x> := PolynomialRing(Rationals()); 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);
 
oscar: Qx, x = PolynomialRing(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)
 

\( x^{28} - x^{27} + 233 x^{26} - 233 x^{25} + 24361 x^{24} - 24361 x^{23} + 1509161 x^{22} + \cdots + 20\!\cdots\!21 \) Copy content Toggle raw display

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

Invariants

Degree:  $28$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(5545505004594863364258565003877268521046795476461201195989861\) \(\medspace = 3^{14}\cdot 11^{14}\cdot 29^{27}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(147.72\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $3^{1/2}11^{1/2}29^{27/28}\approx 147.715635733011$
Ramified primes:   \(3\), \(11\), \(29\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{29}) \)
$\card{ \Gal(K/\Q) }$:  $28$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
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.
Reflex fields:  unavailable$^{8192}$

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

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

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $13$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
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)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
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);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(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);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(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);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A cyclic group of order 28
The 28 conjugacy class representatives for $C_{28}$
Character table for $C_{28}$

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.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

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 ${\href{/padicField/5.7.0.1}{7} }^{4}$ ${\href{/padicField/7.14.0.1}{14} }^{2}$ R ${\href{/padicField/13.7.0.1}{7} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{7}$ $28$ ${\href{/padicField/23.14.0.1}{14} }^{2}$ R $28$ $28$ ${\href{/padicField/41.4.0.1}{4} }^{7}$ $28$ $28$ ${\href{/padicField/53.14.0.1}{14} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{14}$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display Deg $28$$2$$14$$14$
\(11\) Copy content Toggle raw display Deg $28$$2$$14$$14$
\(29\) Copy content Toggle raw display Deg $28$$28$$1$$27$