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Properties

Label 17.17.5447154686...5441.1
Degree $17$
Signature $[17, 0]$
Discriminant $17^{24}\cdot 103^{16}$
Root discriminant $4280.94$
Ramified primes $17, 103$
Class number $17$ (GRH)
Class group $[17]$ (GRH)
Galois group $D_{17}$ (as 17T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![145001801906376687, -11309428987726617, -50160692092741008, 891648638079425, 5600534069106679, 207808365630713, -210262575924428, -15367054046543, 2699830692822, 247158969538, -13391942168, -1455974010, 27140500, 3586048, -21012, -3502, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 3502*x^15 - 21012*x^14 + 3586048*x^13 + 27140500*x^12 - 1455974010*x^11 - 13391942168*x^10 + 247158969538*x^9 + 2699830692822*x^8 - 15367054046543*x^7 - 210262575924428*x^6 + 207808365630713*x^5 + 5600534069106679*x^4 + 891648638079425*x^3 - 50160692092741008*x^2 - 11309428987726617*x + 145001801906376687)
 
gp: K = bnfinit(x^17 - 3502*x^15 - 21012*x^14 + 3586048*x^13 + 27140500*x^12 - 1455974010*x^11 - 13391942168*x^10 + 247158969538*x^9 + 2699830692822*x^8 - 15367054046543*x^7 - 210262575924428*x^6 + 207808365630713*x^5 + 5600534069106679*x^4 + 891648638079425*x^3 - 50160692092741008*x^2 - 11309428987726617*x + 145001801906376687, 1)
 

Normalized defining polynomial

\( x^{17} - 3502 x^{15} - 21012 x^{14} + 3586048 x^{13} + 27140500 x^{12} - 1455974010 x^{11} - 13391942168 x^{10} + 247158969538 x^{9} + 2699830692822 x^{8} - 15367054046543 x^{7} - 210262575924428 x^{6} + 207808365630713 x^{5} + 5600534069106679 x^{4} + 891648638079425 x^{3} - 50160692092741008 x^{2} - 11309428987726617 x + 145001801906376687 \)

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

Invariants

Degree:  $17$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[17, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(54471546860208560987402602575661525149433755592659973376605441=17^{24}\cdot 103^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $4280.94$
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:  $17, 103$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{45} a^{10} + \frac{1}{45} a^{9} + \frac{1}{15} a^{8} + \frac{1}{9} a^{7} + \frac{1}{9} a^{6} + \frac{1}{45} a^{5} + \frac{11}{45} a^{4} - \frac{2}{9} a^{2} - \frac{1}{3} a + \frac{2}{5}$, $\frac{1}{45} a^{11} + \frac{2}{45} a^{9} + \frac{2}{45} a^{8} - \frac{4}{45} a^{6} + \frac{2}{9} a^{5} - \frac{11}{45} a^{4} - \frac{2}{9} a^{3} - \frac{1}{9} a^{2} - \frac{4}{15} a - \frac{2}{5}$, $\frac{1}{45} a^{12} - \frac{2}{15} a^{8} + \frac{1}{45} a^{7} + \frac{17}{45} a^{5} - \frac{2}{45} a^{4} + \frac{2}{9} a^{3} - \frac{7}{45} a^{2} + \frac{4}{15} a + \frac{1}{5}$, $\frac{1}{225} a^{13} - \frac{1}{225} a^{12} - \frac{2}{225} a^{11} + \frac{4}{45} a^{9} - \frac{3}{25} a^{8} + \frac{14}{225} a^{7} - \frac{4}{45} a^{6} + \frac{37}{75} a^{5} - \frac{26}{225} a^{4} - \frac{4}{75} a^{3} + \frac{29}{225} a^{2} + \frac{22}{75} a + \frac{8}{25}$, $\frac{1}{3375} a^{14} - \frac{7}{3375} a^{13} + \frac{14}{3375} a^{12} + \frac{37}{3375} a^{11} - \frac{2}{675} a^{10} - \frac{352}{3375} a^{9} + \frac{526}{3375} a^{8} - \frac{544}{3375} a^{7} - \frac{469}{3375} a^{6} + \frac{223}{3375} a^{5} + \frac{1169}{3375} a^{4} + \frac{1676}{3375} a^{3} - \frac{39}{125} a^{2} + \frac{94}{375} a - \frac{16}{125}$, $\frac{1}{16875} a^{15} + \frac{2}{16875} a^{14} + \frac{26}{16875} a^{13} + \frac{88}{16875} a^{12} + \frac{98}{16875} a^{11} - \frac{67}{16875} a^{10} + \frac{1333}{16875} a^{9} - \frac{272}{3375} a^{8} + \frac{412}{3375} a^{7} + \frac{1177}{16875} a^{6} - \frac{5749}{16875} a^{5} + \frac{2822}{16875} a^{4} + \frac{877}{5625} a^{3} - \frac{277}{5625} a^{2} - \frac{309}{625} a - \frac{144}{625}$, $\frac{1}{3420164651005107976242174131485706443429902671493636348229258523199400363613519210078427912415620260468501050828125} a^{16} - \frac{59301864057608479588155214129161309398109402504469243903048459252070711467110530025863598305005125520570520539}{3420164651005107976242174131485706443429902671493636348229258523199400363613519210078427912415620260468501050828125} a^{15} + \frac{4762859378743631305709004733850287648536824197207334594379659692059830430649978460654512645538157710453567349}{42224254950680345385705853475132178313949415697452300595422944730856794612512582840474418671797780993438284578125} a^{14} - \frac{74791389248521607560182547313724931771238304362693895244418237792906341275041152171839299777792629471276166838}{42224254950680345385705853475132178313949415697452300595422944730856794612512582840474418671797780993438284578125} a^{13} - \frac{4865856299279613945300602117156978009136324846221500244282170411465839697908430764958703684382168106493140366717}{684032930201021595248434826297141288685980534298727269645851704639880072722703842015685582483124052093700210165625} a^{12} - \frac{566007001685041288096395566320192926910443346201658721590195997947952670448619780091629628664332148360945186068}{76003658911224621694270536255237920965108948255414141071761300515542230302522649112853953609236005788188912240625} a^{11} - \frac{17937467450558016853866382653289498257746475699853875822069758691621063151107401700840971717675396204206074953}{228010976733673865082811608765713762895326844766242423215283901546626690907567947338561860827708017364566736721875} a^{10} - \frac{5062602113669242254583701692472524188511975398716619441458687895958270171887740230989606264620251334423794290318}{83418650024514828688833515402090401059265918816917959712908744468278057649110224636059217375990738060207342703125} a^{9} - \frac{4295714773828671315492625585878392528183573668005025960021282824197838373587167287792447754415225330204064227169}{76003658911224621694270536255237920965108948255414141071761300515542230302522649112853953609236005788188912240625} a^{8} + \frac{11578371552390853269457220336862838350218581461494065129296599005194904068919133342388942167017014505374134303338}{380018294556123108471352681276189604825544741277070705358806502577711151512613245564269768046180028940944561203125} a^{7} - \frac{1144290542987826644847326707558322445036691373155886031457043087668678061062615163901813289850163798890171551033}{8403352950872501170128191969252349983857254721114585622184910376411303104701521400684098064903243883214990296875} a^{6} - \frac{85057190170096016120563749037830258727002903371782371732829627427716895578186334047804150976804273148101209517291}{380018294556123108471352681276189604825544741277070705358806502577711151512613245564269768046180028940944561203125} a^{5} - \frac{44487106555716522557136585970305748534744627834737381158374764042160223651751302985783455906050122571546800188308}{92436882459597512871410111661775849822429801932260441844034014140524334151716735407525078713935682715364893265625} a^{4} + \frac{139691058249242896163096374353297396206608603077961614752644775064703711206984653009856183311536819802839693980248}{3420164651005107976242174131485706443429902671493636348229258523199400363613519210078427912415620260468501050828125} a^{3} + \frac{288439206667624103593899111996191804273956882635285542185222404028134208720302093193206417781745922301448166871126}{1140054883668369325414058043828568814476634223831212116076419507733133454537839736692809304138540086822833683609375} a^{2} - \frac{92714524205317572722850708282263228533361739423603780028433906587702585531841939952894210969409063042796242837}{202676423763265657851388096680634455906957195347771042858030134708112614140060397634277209624629348768503765975} a - \frac{56699442897797081306036086830689971718293833877471187368271106998129936832162942681560560305060722134165211592296}{126672764852041036157117560425396534941848247092356901786268834192570383837537748521423256015393342980314853734375}$

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

Class group and class number

$C_{17}$, which has order $17$ (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:  $16$
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:  \( 10720674608800000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{17}$ (as 17T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 34
The 10 conjugacy class representatives for $D_{17}$
Character table for $D_{17}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Galois closure: 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 $17$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $17$ R $17$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $17$ $17$ $17$ $17$

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
17Data not computed
103Data not computed