Properties

Label 28.0.714...129.1
Degree $28$
Signature $[0, 14]$
Discriminant $7.140\times 10^{49}$
Root discriminant $60.32$
Ramified primes $7, 29$
Class number $25984$ (GRH)
Class group $[2, 4, 4, 812]$ (GRH)
Galois group $C_2\times C_{14}$ (as 28T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 + 41*x^26 - 34*x^25 + 814*x^24 - 569*x^23 + 10073*x^22 - 5894*x^21 + 85446*x^20 - 41430*x^19 + 517766*x^18 - 204124*x^17 + 2278084*x^16 - 713493*x^15 + 7270037*x^14 - 1746022*x^13 + 16581844*x^12 - 2930752*x^11 + 26206720*x^10 - 3163168*x^9 + 27305280*x^8 - 2133504*x^7 + 17228288*x^6 - 594944*x^5 + 5748736*x^4 - 229376*x^3 + 745472*x^2 + 57344*x + 16384)
 
gp: K = bnfinit(x^28 - x^27 + 41*x^26 - 34*x^25 + 814*x^24 - 569*x^23 + 10073*x^22 - 5894*x^21 + 85446*x^20 - 41430*x^19 + 517766*x^18 - 204124*x^17 + 2278084*x^16 - 713493*x^15 + 7270037*x^14 - 1746022*x^13 + 16581844*x^12 - 2930752*x^11 + 26206720*x^10 - 3163168*x^9 + 27305280*x^8 - 2133504*x^7 + 17228288*x^6 - 594944*x^5 + 5748736*x^4 - 229376*x^3 + 745472*x^2 + 57344*x + 16384, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16384, 57344, 745472, -229376, 5748736, -594944, 17228288, -2133504, 27305280, -3163168, 26206720, -2930752, 16581844, -1746022, 7270037, -713493, 2278084, -204124, 517766, -41430, 85446, -5894, 10073, -569, 814, -34, 41, -1, 1]);
 

\(x^{28} - x^{27} + 41 x^{26} - 34 x^{25} + 814 x^{24} - 569 x^{23} + 10073 x^{22} - 5894 x^{21} + 85446 x^{20} - 41430 x^{19} + 517766 x^{18} - 204124 x^{17} + 2278084 x^{16} - 713493 x^{15} + 7270037 x^{14} - 1746022 x^{13} + 16581844 x^{12} - 2930752 x^{11} + 26206720 x^{10} - 3163168 x^{9} + 27305280 x^{8} - 2133504 x^{7} + 17228288 x^{6} - 594944 x^{5} + 5748736 x^{4} - 229376 x^{3} + 745472 x^{2} + 57344 x + 16384\)  Toggle raw display

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

Invariants

Degree:  $28$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(71403665296191917297019015087709113341743884283129\)\(\medspace = 7^{14}\cdot 29^{26}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $60.32$
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$$\chi_{203}(64,·)$, $\chi_{203}(1,·)$, $\chi_{203}(132,·)$, $\chi_{203}(197,·)$, $\chi_{203}(6,·)$, $\chi_{203}(71,·)$, $\chi_{203}(202,·)$, $\chi_{203}(139,·)$, $\chi_{203}(13,·)$, $\chi_{203}(78,·)$, $\chi_{203}(141,·)$, $\chi_{203}(120,·)$, $\chi_{203}(146,·)$, $\chi_{203}(83,·)$, $\chi_{203}(20,·)$, $\chi_{203}(22,·)$, $\chi_{203}(92,·)$, $\chi_{203}(34,·)$, $\chi_{203}(36,·)$, $\chi_{203}(167,·)$, $\chi_{203}(169,·)$, $\chi_{203}(111,·)$, $\chi_{203}(181,·)$, $\chi_{203}(183,·)$, $\chi_{203}(62,·)$, $\chi_{203}(57,·)$, $\chi_{203}(125,·)$, $\chi_{203}(190,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{15} + \frac{1}{4} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{16} + \frac{1}{8} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{8} a^{12} + \frac{1}{8} a^{11} + \frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{3}{8} a^{4} - \frac{3}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{18} - \frac{1}{16} a^{17} + \frac{1}{16} a^{16} - \frac{1}{8} a^{15} - \frac{1}{8} a^{14} - \frac{1}{16} a^{13} - \frac{7}{16} a^{12} + \frac{1}{8} a^{11} + \frac{3}{8} a^{10} + \frac{1}{8} a^{9} + \frac{3}{8} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{5}{16} a^{5} + \frac{5}{16} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{19} - \frac{1}{32} a^{18} + \frac{1}{32} a^{17} - \frac{1}{16} a^{16} - \frac{1}{16} a^{15} - \frac{1}{32} a^{14} + \frac{9}{32} a^{13} - \frac{7}{16} a^{12} + \frac{3}{16} a^{11} + \frac{1}{16} a^{10} - \frac{5}{16} a^{9} + \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{5}{32} a^{6} - \frac{11}{32} a^{5} + \frac{1}{16} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{64} a^{20} - \frac{1}{64} a^{19} + \frac{1}{64} a^{18} - \frac{1}{32} a^{17} - \frac{1}{32} a^{16} - \frac{1}{64} a^{15} + \frac{9}{64} a^{14} - \frac{7}{32} a^{13} + \frac{3}{32} a^{12} - \frac{15}{32} a^{11} + \frac{11}{32} a^{10} - \frac{7}{16} a^{9} - \frac{7}{16} a^{8} - \frac{5}{64} a^{7} - \frac{11}{64} a^{6} + \frac{1}{32} a^{5} - \frac{3}{16} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{128} a^{21} - \frac{1}{128} a^{20} + \frac{1}{128} a^{19} - \frac{1}{64} a^{18} - \frac{1}{64} a^{17} - \frac{1}{128} a^{16} + \frac{9}{128} a^{15} - \frac{7}{64} a^{14} + \frac{3}{64} a^{13} - \frac{15}{64} a^{12} + \frac{11}{64} a^{11} + \frac{9}{32} a^{10} - \frac{7}{32} a^{9} + \frac{59}{128} a^{8} + \frac{53}{128} a^{7} - \frac{31}{64} a^{6} - \frac{3}{32} a^{5} + \frac{3}{16} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{256} a^{22} - \frac{1}{256} a^{21} + \frac{1}{256} a^{20} - \frac{1}{128} a^{19} - \frac{1}{128} a^{18} - \frac{1}{256} a^{17} + \frac{9}{256} a^{16} - \frac{7}{128} a^{15} + \frac{3}{128} a^{14} - \frac{15}{128} a^{13} + \frac{11}{128} a^{12} - \frac{23}{64} a^{11} - \frac{7}{64} a^{10} + \frac{59}{256} a^{9} - \frac{75}{256} a^{8} - \frac{31}{128} a^{7} + \frac{29}{64} a^{6} - \frac{13}{32} a^{5} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{512} a^{23} - \frac{1}{512} a^{22} + \frac{1}{512} a^{21} - \frac{1}{256} a^{20} - \frac{1}{256} a^{19} - \frac{1}{512} a^{18} + \frac{9}{512} a^{17} - \frac{7}{256} a^{16} + \frac{3}{256} a^{15} - \frac{15}{256} a^{14} + \frac{11}{256} a^{13} - \frac{23}{128} a^{12} - \frac{7}{128} a^{11} - \frac{197}{512} a^{10} + \frac{181}{512} a^{9} + \frac{97}{256} a^{8} - \frac{35}{128} a^{7} - \frac{13}{64} a^{6} + \frac{1}{16} a^{5} - \frac{1}{4} a^{4} - \frac{3}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{1024} a^{24} - \frac{1}{1024} a^{23} + \frac{1}{1024} a^{22} - \frac{1}{512} a^{21} - \frac{1}{512} a^{20} - \frac{1}{1024} a^{19} + \frac{9}{1024} a^{18} - \frac{7}{512} a^{17} + \frac{3}{512} a^{16} - \frac{15}{512} a^{15} + \frac{11}{512} a^{14} - \frac{23}{256} a^{13} - \frac{7}{256} a^{12} - \frac{197}{1024} a^{11} - \frac{331}{1024} a^{10} - \frac{159}{512} a^{9} - \frac{35}{256} a^{8} + \frac{51}{128} a^{7} + \frac{1}{32} a^{6} - \frac{1}{8} a^{5} + \frac{5}{16} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{2048} a^{25} - \frac{1}{2048} a^{24} + \frac{1}{2048} a^{23} - \frac{1}{1024} a^{22} - \frac{1}{1024} a^{21} - \frac{1}{2048} a^{20} + \frac{9}{2048} a^{19} - \frac{7}{1024} a^{18} + \frac{3}{1024} a^{17} - \frac{15}{1024} a^{16} + \frac{11}{1024} a^{15} - \frac{23}{512} a^{14} - \frac{7}{512} a^{13} - \frac{197}{2048} a^{12} - \frac{331}{2048} a^{11} - \frac{159}{1024} a^{10} - \frac{35}{512} a^{9} + \frac{51}{256} a^{8} - \frac{31}{64} a^{7} + \frac{7}{16} a^{6} - \frac{11}{32} a^{5} + \frac{1}{16} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4096} a^{26} - \frac{1}{4096} a^{25} + \frac{1}{4096} a^{24} - \frac{1}{2048} a^{23} - \frac{1}{2048} a^{22} - \frac{1}{4096} a^{21} + \frac{9}{4096} a^{20} - \frac{7}{2048} a^{19} + \frac{3}{2048} a^{18} - \frac{15}{2048} a^{17} + \frac{11}{2048} a^{16} - \frac{23}{1024} a^{15} - \frac{7}{1024} a^{14} - \frac{197}{4096} a^{13} + \frac{1717}{4096} a^{12} + \frac{865}{2048} a^{11} - \frac{35}{1024} a^{10} + \frac{51}{512} a^{9} - \frac{31}{128} a^{8} - \frac{9}{32} a^{7} + \frac{21}{64} a^{6} + \frac{1}{32} a^{5} + \frac{5}{16} a^{4} - \frac{1}{8} a^{3}$, $\frac{1}{154775751925570245573958597181841542237250043793935653191765614592} a^{27} + \frac{5162156017537702077435326049760612207367162059729754419676353}{154775751925570245573958597181841542237250043793935653191765614592} a^{26} + \frac{18656683733351289823085495363467723786329677160696553795874535}{154775751925570245573958597181841542237250043793935653191765614592} a^{25} - \frac{312616077549038492401224942781122999041453407838210665471639}{19346968990696280696744824647730192779656255474241956648970701824} a^{24} - \frac{14531844092603560289786965546375805353511075693511092536101895}{77387875962785122786979298590920771118625021896967826595882807296} a^{23} - \frac{45389502053735659888696892935542232400810849982731104596036709}{154775751925570245573958597181841542237250043793935653191765614592} a^{22} - \frac{375526012514034101600943074910150579024848777439816405842968425}{154775751925570245573958597181841542237250043793935653191765614592} a^{21} - \frac{296370883385518533012628517494792781778465621945290886826006353}{38693937981392561393489649295460385559312510948483913297941403648} a^{20} - \frac{678383559875014800881098380784271619544819169201631615338388111}{77387875962785122786979298590920771118625021896967826595882807296} a^{19} - \frac{75417652553018589397441662347702182524765972257451658942231049}{77387875962785122786979298590920771118625021896967826595882807296} a^{18} + \frac{607883488731163533169395699787034117422762499875488953771219573}{77387875962785122786979298590920771118625021896967826595882807296} a^{17} - \frac{191550642535794307056949870417341961211404765689013330185542853}{2418371123837035087093103080966274097457031934280244581121337728} a^{16} + \frac{8796044832763539001644356983729082980292766951408578682290826735}{38693937981392561393489649295460385559312510948483913297941403648} a^{15} - \frac{3684559652302240976761110447309422285632194451091198988345717613}{9104455995621779151409329245990678955132355517290332540692094976} a^{14} + \frac{77220887619141692918796874438319466420277633682715165847916316363}{154775751925570245573958597181841542237250043793935653191765614592} a^{13} - \frac{15268638684844896203577132550108441727831798573854293774261392103}{38693937981392561393489649295460385559312510948483913297941403648} a^{12} + \frac{41832292319915189562393665400255879933956229004384486495123533}{569028499726361196963083077874417434695772219830645783793255936} a^{11} - \frac{577008272604624122135361607110899973753108911305224086884416213}{2418371123837035087093103080966274097457031934280244581121337728} a^{10} - \frac{691388895604299065914527806319741484508932927067455024394737511}{9673484495348140348372412323865096389828127737120978324485350912} a^{9} - \frac{383589786102558078933288694904273464893461071426249084773517759}{2418371123837035087093103080966274097457031934280244581121337728} a^{8} - \frac{278280619192175100609111938067448987488552779616813196917803445}{1209185561918517543546551540483137048728515967140122290560668864} a^{7} - \frac{40773786795272231845171416729866428584786063513657076343535455}{1209185561918517543546551540483137048728515967140122290560668864} a^{6} - \frac{46866805316658691036107717675718650691504770866047904162595241}{151148195239814692943318942560392131091064495892515286320083608} a^{5} - \frac{132033474021269783018197837332564523370420946162915324556114317}{302296390479629385886637885120784262182128991785030572640167216} a^{4} - \frac{75236580563295033267179064801429563396391262192404212985861993}{151148195239814692943318942560392131091064495892515286320083608} a^{3} - \frac{13247145502936068231133112101220515780412789544337767621115267}{75574097619907346471659471280196065545532247946257643160041804} a^{2} - \frac{15946298763610656873966970533432879401827224861659141913031995}{37787048809953673235829735640098032772766123973128821580020902} a + \frac{1110748474108329495595811321531959134333241441007588272253588}{18893524404976836617914867820049016386383061986564410790010451}$  Toggle raw display

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{812}$, which has order $25984$ (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:  $13$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
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:  \( 487075979.1876791 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{14}\cdot 487075979.1876791 \cdot 25984}{2\sqrt{71403665296191917297019015087709113341743884283129}}\approx 0.111925908545134$ (assuming GRH)

Galois group

$C_2\times C_{14}$ (as 28T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
An abelian group of order 28
The 28 conjugacy class representatives for $C_2\times C_{14}$
Character table for $C_2\times C_{14}$ is not computed

Intermediate fields

\(\Q(\sqrt{-203}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-7}, \sqrt{29})\), 7.7.594823321.1, 14.0.8450068952156066122535627.1, \(\Q(\zeta_{29})^+\), 14.0.291381688005381590432263.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{4}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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}$
$29$29.14.13.1$x^{14} - 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$
29.14.13.1$x^{14} - 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$