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Properties

Label 28.0.12984931501...0096.1
Degree $28$
Signature $[0, 14]$
Discriminant $2^{42}\cdot 43^{26}$
Root discriminant $92.97$
Ramified primes $2, 43$
Class number $203$ (GRH)
Class group $[203]$ (GRH)
Galois group $C_2\times C_{14}$ (as 28T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9978409, -40938072, 75048342, -88401976, 99986627, -114631858, 107446012, -89019544, 82171671, -67452498, 44836479, -26733774, 15305413, -6552716, 3203837, -1214276, 638481, -200258, 139895, -20814, 27349, -84, 3692, 232, 400, -8, 33, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 2*x^27 + 33*x^26 - 8*x^25 + 400*x^24 + 232*x^23 + 3692*x^22 - 84*x^21 + 27349*x^20 - 20814*x^19 + 139895*x^18 - 200258*x^17 + 638481*x^16 - 1214276*x^15 + 3203837*x^14 - 6552716*x^13 + 15305413*x^12 - 26733774*x^11 + 44836479*x^10 - 67452498*x^9 + 82171671*x^8 - 89019544*x^7 + 107446012*x^6 - 114631858*x^5 + 99986627*x^4 - 88401976*x^3 + 75048342*x^2 - 40938072*x + 9978409)
 
gp: K = bnfinit(x^28 - 2*x^27 + 33*x^26 - 8*x^25 + 400*x^24 + 232*x^23 + 3692*x^22 - 84*x^21 + 27349*x^20 - 20814*x^19 + 139895*x^18 - 200258*x^17 + 638481*x^16 - 1214276*x^15 + 3203837*x^14 - 6552716*x^13 + 15305413*x^12 - 26733774*x^11 + 44836479*x^10 - 67452498*x^9 + 82171671*x^8 - 89019544*x^7 + 107446012*x^6 - 114631858*x^5 + 99986627*x^4 - 88401976*x^3 + 75048342*x^2 - 40938072*x + 9978409, 1)
 

Normalized defining polynomial

\( x^{28} - 2 x^{27} + 33 x^{26} - 8 x^{25} + 400 x^{24} + 232 x^{23} + 3692 x^{22} - 84 x^{21} + 27349 x^{20} - 20814 x^{19} + 139895 x^{18} - 200258 x^{17} + 638481 x^{16} - 1214276 x^{15} + 3203837 x^{14} - 6552716 x^{13} + 15305413 x^{12} - 26733774 x^{11} + 44836479 x^{10} - 67452498 x^{9} + 82171671 x^{8} - 89019544 x^{7} + 107446012 x^{6} - 114631858 x^{5} + 99986627 x^{4} - 88401976 x^{3} + 75048342 x^{2} - 40938072 x + 9978409 \)

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:  \(12984931501152231903148118403201067499769297367881220096=2^{42}\cdot 43^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $92.97$
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:  $2, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(344=2^{3}\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{344}(1,·)$, $\chi_{344}(131,·)$, $\chi_{344}(211,·)$, $\chi_{344}(257,·)$, $\chi_{344}(137,·)$, $\chi_{344}(11,·)$, $\chi_{344}(145,·)$, $\chi_{344}(75,·)$, $\chi_{344}(259,·)$, $\chi_{344}(217,·)$, $\chi_{344}(299,·)$, $\chi_{344}(27,·)$, $\chi_{344}(323,·)$, $\chi_{344}(161,·)$, $\chi_{344}(35,·)$, $\chi_{344}(113,·)$, $\chi_{344}(65,·)$, $\chi_{344}(97,·)$, $\chi_{344}(41,·)$, $\chi_{344}(193,·)$, $\chi_{344}(171,·)$, $\chi_{344}(305,·)$, $\chi_{344}(51,·)$, $\chi_{344}(219,·)$, $\chi_{344}(297,·)$, $\chi_{344}(121,·)$, $\chi_{344}(59,·)$, $\chi_{344}(107,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{7} a^{21} - \frac{2}{7} a^{19} - \frac{2}{7} a^{18} + \frac{1}{7} a^{17} - \frac{2}{7} a^{16} + \frac{2}{7} a^{15} - \frac{3}{7} a^{14} - \frac{2}{7} a^{13} - \frac{3}{7} a^{10} + \frac{3}{7} a^{8} - \frac{2}{7} a^{7} + \frac{3}{7} a^{6} + \frac{2}{7} a^{5} - \frac{3}{7} a^{4} + \frac{3}{7} a^{2} + \frac{3}{7} a$, $\frac{1}{7} a^{22} - \frac{2}{7} a^{20} - \frac{2}{7} a^{19} + \frac{1}{7} a^{18} - \frac{2}{7} a^{17} + \frac{2}{7} a^{16} - \frac{3}{7} a^{15} - \frac{2}{7} a^{14} - \frac{3}{7} a^{11} + \frac{3}{7} a^{9} - \frac{2}{7} a^{8} + \frac{3}{7} a^{7} + \frac{2}{7} a^{6} - \frac{3}{7} a^{5} + \frac{3}{7} a^{3} + \frac{3}{7} a^{2}$, $\frac{1}{7} a^{23} - \frac{2}{7} a^{20} - \frac{3}{7} a^{19} + \frac{1}{7} a^{18} - \frac{3}{7} a^{17} + \frac{2}{7} a^{15} + \frac{1}{7} a^{14} + \frac{3}{7} a^{13} - \frac{3}{7} a^{12} - \frac{3}{7} a^{10} - \frac{2}{7} a^{9} + \frac{2}{7} a^{8} - \frac{2}{7} a^{7} + \frac{3}{7} a^{6} - \frac{3}{7} a^{5} - \frac{3}{7} a^{4} + \frac{3}{7} a^{3} - \frac{1}{7} a^{2} - \frac{1}{7} a$, $\frac{1}{2149} a^{24} + \frac{51}{2149} a^{23} + \frac{149}{2149} a^{22} + \frac{143}{2149} a^{21} - \frac{592}{2149} a^{20} + \frac{1038}{2149} a^{19} + \frac{810}{2149} a^{18} + \frac{828}{2149} a^{17} - \frac{816}{2149} a^{16} - \frac{523}{2149} a^{15} - \frac{110}{307} a^{14} + \frac{20}{307} a^{13} - \frac{916}{2149} a^{12} - \frac{289}{2149} a^{11} + \frac{1013}{2149} a^{10} + \frac{655}{2149} a^{9} - \frac{596}{2149} a^{8} + \frac{79}{2149} a^{7} + \frac{470}{2149} a^{6} - \frac{26}{2149} a^{5} + \frac{787}{2149} a^{4} - \frac{115}{2149} a^{3} - \frac{1053}{2149} a^{2} + \frac{1014}{2149} a + \frac{111}{307}$, $\frac{1}{539399} a^{25} + \frac{111}{539399} a^{24} - \frac{28719}{539399} a^{23} + \frac{16451}{539399} a^{22} + \frac{2720}{77057} a^{21} - \frac{77769}{539399} a^{20} - \frac{193869}{539399} a^{19} + \frac{24254}{539399} a^{18} - \frac{104022}{539399} a^{17} + \frac{104324}{539399} a^{16} - \frac{216350}{539399} a^{15} + \frac{54636}{539399} a^{14} + \frac{3876}{77057} a^{13} - \frac{109895}{539399} a^{12} - \frac{124391}{539399} a^{11} - \frac{267055}{539399} a^{10} + \frac{184836}{539399} a^{9} - \frac{115501}{539399} a^{8} + \frac{228092}{539399} a^{7} - \frac{178744}{539399} a^{6} - \frac{29407}{77057} a^{5} - \frac{242703}{539399} a^{4} + \frac{77393}{539399} a^{3} + \frac{63704}{539399} a^{2} + \frac{120254}{539399} a - \frac{10532}{77057}$, $\frac{1}{112356113433738723829} a^{26} - \frac{10359394090574}{112356113433738723829} a^{25} - \frac{9047342130086614}{112356113433738723829} a^{24} + \frac{5817373332292205616}{112356113433738723829} a^{23} + \frac{397413501679815900}{16050873347676960547} a^{22} - \frac{7141301278949907780}{112356113433738723829} a^{21} + \frac{5653053982027195853}{16050873347676960547} a^{20} - \frac{1273651828739535232}{16050873347676960547} a^{19} + \frac{4054140508274638270}{16050873347676960547} a^{18} - \frac{5642214986579008389}{112356113433738723829} a^{17} + \frac{33069829990652481162}{112356113433738723829} a^{16} - \frac{41027478842135372192}{112356113433738723829} a^{15} + \frac{4853731259232520915}{112356113433738723829} a^{14} - \frac{19101148580237065103}{112356113433738723829} a^{13} - \frac{8169524865759147417}{112356113433738723829} a^{12} + \frac{21149607137895429580}{112356113433738723829} a^{11} - \frac{55128853951174234765}{112356113433738723829} a^{10} + \frac{24594355621413152681}{112356113433738723829} a^{9} - \frac{7860092163288248990}{112356113433738723829} a^{8} + \frac{40156325040528028972}{112356113433738723829} a^{7} + \frac{52757076617737194965}{112356113433738723829} a^{6} - \frac{43049720927374221444}{112356113433738723829} a^{5} - \frac{56012159354501531099}{112356113433738723829} a^{4} + \frac{54779959282789104936}{112356113433738723829} a^{3} - \frac{54773339417418601804}{112356113433738723829} a^{2} - \frac{12610502410050600806}{112356113433738723829} a + \frac{6315539909365142127}{16050873347676960547}$, $\frac{1}{451883458366208575486254559550926630614816528139230088477367485834297143071588463853} a^{27} + \frac{1849002133060682708125159040452806315068054452893451045955473202}{451883458366208575486254559550926630614816528139230088477367485834297143071588463853} a^{26} - \frac{152936659338559829305422876793657458142611170119000259882549142350958926567512}{451883458366208575486254559550926630614816528139230088477367485834297143071588463853} a^{25} - \frac{84047948448581132330359882975782975405341878587032417876971798763632948787646305}{451883458366208575486254559550926630614816528139230088477367485834297143071588463853} a^{24} - \frac{13501347310322174607251794299082740704036095990171210564708722890891327145830196526}{451883458366208575486254559550926630614816528139230088477367485834297143071588463853} a^{23} + \frac{4833717706252262297471023071227867738107057890584242815567984198244611371766710294}{451883458366208575486254559550926630614816528139230088477367485834297143071588463853} a^{22} - \frac{4869149425370110009910448694920693324381788698514553461919091126627445721112442902}{451883458366208575486254559550926630614816528139230088477367485834297143071588463853} a^{21} + \frac{143875974760404670456971626882912552857463699030938246947064455859443850106554961192}{451883458366208575486254559550926630614816528139230088477367485834297143071588463853} a^{20} + \frac{17670390862742504425026054334485562385044024175054157467803016017997861909206697048}{64554779766601225069464937078703804373545218305604298353909640833471020438798351979} a^{19} - \frac{218674839670015111245936271985063964063840834040722532767929974875344761856685362166}{451883458366208575486254559550926630614816528139230088477367485834297143071588463853} a^{18} + \frac{8098217644007054762800866950996242329217440774565630453441019112831349026445658972}{451883458366208575486254559550926630614816528139230088477367485834297143071588463853} a^{17} + \frac{63721608721347448495143551171494505511415873157153033852633908040162368634335172581}{451883458366208575486254559550926630614816528139230088477367485834297143071588463853} a^{16} + \frac{71589903191625227662008845434843237783203643624516368827056878032547646933193488775}{451883458366208575486254559550926630614816528139230088477367485834297143071588463853} a^{15} + \frac{32593925665728238414684413303394927767355954707326787581685522216260178944637467977}{451883458366208575486254559550926630614816528139230088477367485834297143071588463853} a^{14} + \frac{101639160862749700126041006651754159527395049375845638012044286472303868299538927880}{451883458366208575486254559550926630614816528139230088477367485834297143071588463853} a^{13} - \frac{16756862006836497037542693822810514112814917032894538355399530680441814556489678455}{451883458366208575486254559550926630614816528139230088477367485834297143071588463853} a^{12} - \frac{166012250270799693581081904615501611584358499518409796016133805116172320078211702723}{451883458366208575486254559550926630614816528139230088477367485834297143071588463853} a^{11} + \frac{2389740456086119458167209387019069418156052480127929482397017762549678834344939126}{5720043776787450322610817209505400387529323141002912512371740327016419532551752707} a^{10} + \frac{1419133662893562225884691277109131911193988460943091217362199968147159994786603932}{64554779766601225069464937078703804373545218305604298353909640833471020438798351979} a^{9} + \frac{1791758797673543325313759203492810088664099557925593920625275718146247426486844170}{64554779766601225069464937078703804373545218305604298353909640833471020438798351979} a^{8} + \frac{19071554211811682717578076538463567466773134382870678014567069091832815623909912913}{451883458366208575486254559550926630614816528139230088477367485834297143071588463853} a^{7} + \frac{153714359834389529594021208836669807685591031849973064044117360572027248053788244713}{451883458366208575486254559550926630614816528139230088477367485834297143071588463853} a^{6} - \frac{156345173953036984147431493942444097927179499011619569868131410023406585905823645458}{451883458366208575486254559550926630614816528139230088477367485834297143071588463853} a^{5} + \frac{179378669366755473454623231514397374358063462711420527576281220285582588787710215931}{451883458366208575486254559550926630614816528139230088477367485834297143071588463853} a^{4} + \frac{160389759719195357819377599313872302661429242958210458053727159103193622576233430951}{451883458366208575486254559550926630614816528139230088477367485834297143071588463853} a^{3} - \frac{208488772374403936132942175825700148564437291454635935444042314454954403043580838851}{451883458366208575486254559550926630614816528139230088477367485834297143071588463853} a^{2} + \frac{149419883126462778662343165592748436412001727275635686359991664985414644576454775280}{451883458366208575486254559550926630614816528139230088477367485834297143071588463853} a + \frac{27064154613324125077172526337762033838214219210997527759205464110951679731219927952}{64554779766601225069464937078703804373545218305604298353909640833471020438798351979}$

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

Class group and class number

$C_{203}$, which has order $203$ (assuming GRH)

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:  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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 40040306449331.87 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
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

\(\Q(\sqrt{86}) \), \(\Q(\sqrt{-43}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-2}, \sqrt{-43})\), 7.7.6321363049.1, 14.14.3603461044766854684853927936.1, 14.0.1718264124282290785243.1, 14.0.83801419645740806624509952.1

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 R ${\href{/LocalNumberField/3.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{4}$ R ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{4}$

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
2Data not computed
$43$43.14.13.11$x^{14} + 205667667$$14$$1$$13$$C_{14}$$[\ ]_{14}$
43.14.13.11$x^{14} + 205667667$$14$$1$$13$$C_{14}$$[\ ]_{14}$