↑

Properties

Label 14.14.2443128488...1616.1
Degree $14$
Signature $[14, 0]$
Discriminant $2^{36}\cdot 3^{16}\cdot 7^{18}\cdot 4145023^{6}$
Root discriminant $174{,}580.67$
Ramified primes $2, 3, 7, 4145023$
Class number Not computed
Class group Not computed
Galois group 14T22

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-83173498199506854751458701376, 77844118533852728698751040, 60126791399546070679893112, -39871919309692447523616, -16633213695413438344032, 7743078094604088768, 2275891736459084940, -752915315481312, -166594899026864, 33352434192, 6567239322, -517832, -129864, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 129864*x^12 - 517832*x^11 + 6567239322*x^10 + 33352434192*x^9 - 166594899026864*x^8 - 752915315481312*x^7 + 2275891736459084940*x^6 + 7743078094604088768*x^5 - 16633213695413438344032*x^4 - 39871919309692447523616*x^3 + 60126791399546070679893112*x^2 + 77844118533852728698751040*x - 83173498199506854751458701376)
 
gp: K = bnfinit(x^14 - 129864*x^12 - 517832*x^11 + 6567239322*x^10 + 33352434192*x^9 - 166594899026864*x^8 - 752915315481312*x^7 + 2275891736459084940*x^6 + 7743078094604088768*x^5 - 16633213695413438344032*x^4 - 39871919309692447523616*x^3 + 60126791399546070679893112*x^2 + 77844118533852728698751040*x - 83173498199506854751458701376, 1)
 

Normalized defining polynomial

\( x^{14} - 129864 x^{12} - 517832 x^{11} + 6567239322 x^{10} + 33352434192 x^{9} - 166594899026864 x^{8} - 752915315481312 x^{7} + 2275891736459084940 x^{6} + 7743078094604088768 x^{5} - 16633213695413438344032 x^{4} - 39871919309692447523616 x^{3} + 60126791399546070679893112 x^{2} + 77844118533852728698751040 x - 83173498199506854751458701376 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(24431284889171154824543220676517883413839621617385499051692974999513071616=2^{36}\cdot 3^{16}\cdot 7^{18}\cdot 4145023^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $174{,}580.67$
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:  $2, 3, 7, 4145023$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{4} a^{5} - \frac{1}{2} a$, $\frac{1}{16} a^{6} - \frac{1}{2} a^{3} - \frac{1}{8} a^{2}$, $\frac{1}{224} a^{7} + \frac{3}{112} a^{6} + \frac{1}{14} a^{5} + \frac{1}{28} a^{4} + \frac{55}{112} a^{3} + \frac{25}{56} a^{2} + \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{224} a^{8} - \frac{3}{112} a^{6} + \frac{3}{28} a^{5} - \frac{25}{112} a^{4} + \frac{27}{56} a^{2} + \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{224} a^{9} + \frac{1}{56} a^{6} - \frac{5}{112} a^{5} + \frac{3}{14} a^{4} + \frac{3}{7} a^{3} + \frac{9}{28} a^{2} + \frac{1}{14} a + \frac{1}{7}$, $\frac{1}{3584} a^{10} - \frac{1}{448} a^{9} - \frac{1}{448} a^{8} - \frac{1}{896} a^{6} + \frac{3}{112} a^{5} - \frac{11}{56} a^{4} + \frac{3}{14} a^{3} + \frac{321}{896} a^{2} + \frac{51}{112} a - \frac{31}{112}$, $\frac{1}{3584} a^{11} - \frac{1}{448} a^{9} - \frac{1}{896} a^{7} - \frac{1}{56} a^{6} + \frac{1}{56} a^{5} + \frac{3}{28} a^{4} - \frac{191}{896} a^{3} - \frac{13}{28} a^{2} - \frac{31}{112} a - \frac{3}{14}$, $\frac{1}{25088} a^{12} + \frac{1}{12544} a^{11} + \frac{3}{25088} a^{10} - \frac{3}{3136} a^{9} - \frac{11}{6272} a^{8} - \frac{1}{3136} a^{7} - \frac{13}{896} a^{6} - \frac{3}{49} a^{5} - \frac{551}{6272} a^{4} - \frac{863}{3136} a^{3} - \frac{45}{6272} a^{2} - \frac{149}{784} a + \frac{363}{784}$, $\frac{1}{58035620484733436848994259157221816964672319363627143600875331496923026412863533965059225362374886106323817446089092391123180618752} a^{13} + \frac{9978127507910464900813028066849306297129034202311879637971630161329014747232384190583421720522713139274020131588497689849981}{690900243865874248202312609014545440055622849566989804772325374963369362057899213869752682885415310789569255310584433227656912128} a^{12} + \frac{476287496222982601924418760125468965350014726383962614719768459356729430669797344760683738844143677830253585113614849348465543}{9672603414122239474832376526203636160778719893937857266812555249487171068810588994176537560395814351053969574348182065187196769792} a^{11} - \frac{1399468222037840056982556716033805498308471930060422894788113852078557419344885349750438340777586752765123513128336225905108671}{29017810242366718424497129578610908482336159681813571800437665748461513206431766982529612681187443053161908723044546195561590309376} a^{10} + \frac{68867000361573420100664816076224800623231330400752787536405427228710519100196217512446166731707287712911453571070827365005677}{4836301707061119737416188263101818080389359946968928633406277624743585534405294497088268780197907175526984787174091032593598384896} a^{9} - \frac{516624245051290810606680079659078841987782620652723140891113073736854170396813919521458966371200789715560211510047260436082201}{604537713382639967177023532887727260048669993371116079175784703092948191800661812136033597524738396940873098396761379074199798112} a^{8} - \frac{10520958465561648495773609527643533028202239826336313264033811783025127357396872229848649112318690597579799700531382818055305753}{7254452560591679606124282394652727120584039920453392950109416437115378301607941745632403170296860763290477180761136548890397577344} a^{7} - \frac{40999965365969661502218510109953373720022614529650825276085529443123944378406334569303197152024709430280419585222130620481710795}{2418150853530559868708094131550909040194679973484464316703138812371792767202647248544134390098953587763492393587045516296799192448} a^{6} + \frac{66559707734956677607476514460546739590272382795030452890262431569500330526609418179235207655243109178839013715486217170241000685}{690900243865874248202312609014545440055622849566989804772325374963369362057899213869752682885415310789569255310584433227656912128} a^{5} + \frac{36798353095553841299054998124383958714573474026792616306290448605709903781604956484160420474623471649269812117284548255564292507}{1209075426765279934354047065775454520097339986742232158351569406185896383601323624272067195049476793881746196793522758148399596224} a^{4} - \frac{609611818127388100863643454590861455397159049930620145305073418761369320499306954255533087247571861371899798841103667907860168369}{2418150853530559868708094131550909040194679973484464316703138812371792767202647248544134390098953587763492393587045516296799192448} a^{3} - \frac{681580085790077546087470505135840825649250427978463369326251234967953651018872606863513163130340733421968414388683538143104524885}{2418150853530559868708094131550909040194679973484464316703138812371792767202647248544134390098953587763492393587045516296799192448} a^{2} + \frac{4816960087423641906800365378524263521235390001304524777954569884135305531980109330679556534504638331699646708267906524745363925}{226701642518489987691383824832897722518251247514168529690919263659855571925248179551012599071776898852827411898785517152824924292} a - \frac{16777809599469699109504236819481343239152869631985206987319044487008476254955828824091193225639279141803423794470002390307482093}{302268856691319983588511766443863630024334996685558039587892351546474095900330906068016798762369198470436549198380689537099899056}$

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

14T22:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 588
The 10 conjugacy class representatives for [1/6_-.F_42(7)^2]2_2
Character table for [1/6_-.F_42(7)^2]2_2

Intermediate fields

\(\Q(\sqrt{2}) \)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 28 sibling: data not computed
Degree 42 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$2$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.12.16.30$x^{12} + 93 x^{11} + 351 x^{10} + 3 x^{9} + 126 x^{8} - 297 x^{7} + 171 x^{6} + 243 x^{5} - 324 x^{4} - 54 x^{3} + 162 x^{2} - 243 x + 324$$3$$4$$16$$C_3 : C_4$$[2]^{4}$
$7$7.7.10.4$x^{7} + 21 x^{4} + 7$$7$$1$$10$$F_7$$[5/3]_{3}^{2}$
7.7.8.6$x^{7} + 35 x^{2} + 7$$7$$1$$8$$F_7$$[4/3]_{3}^{2}$
4145023Data not computed