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Properties

Label 17.1.39726964294...6976.1
Degree $17$
Signature $[1, 8]$
Discriminant $2^{24}\cdot 17^{32}$
Root discriminant $550.96$
Ramified primes $2, 17$
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![1767616042808, 2281813189964, 138506493022, -328446625597, 217180838276, 12487240571, -45700927246, 12939379031, -1851117488, 159332007, -5536254, 2452233, -97036, 19601, -850, 221, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^17 + 221*x^15 - 850*x^14 + 19601*x^13 - 97036*x^12 + 2452233*x^11 - 5536254*x^10 + 159332007*x^9 - 1851117488*x^8 + 12939379031*x^7 - 45700927246*x^6 + 12487240571*x^5 + 217180838276*x^4 - 328446625597*x^3 + 138506493022*x^2 + 2281813189964*x + 1767616042808)
 
gp: K = bnfinit(x^17 + 221*x^15 - 850*x^14 + 19601*x^13 - 97036*x^12 + 2452233*x^11 - 5536254*x^10 + 159332007*x^9 - 1851117488*x^8 + 12939379031*x^7 - 45700927246*x^6 + 12487240571*x^5 + 217180838276*x^4 - 328446625597*x^3 + 138506493022*x^2 + 2281813189964*x + 1767616042808, 1)
 

Normalized defining polynomial

\( x^{17} + 221 x^{15} - 850 x^{14} + 19601 x^{13} - 97036 x^{12} + 2452233 x^{11} - 5536254 x^{10} + 159332007 x^{9} - 1851117488 x^{8} + 12939379031 x^{7} - 45700927246 x^{6} + 12487240571 x^{5} + 217180838276 x^{4} - 328446625597 x^{3} + 138506493022 x^{2} + 2281813189964 x + 1767616042808 \)

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

Invariants

Degree:  $17$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(39726964294200827244451673677113820402444926976=2^{24}\cdot 17^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $550.96$
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:  $2, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{176} a^{9} + \frac{3}{88} a^{8} - \frac{5}{176} a^{7} + \frac{5}{88} a^{6} - \frac{21}{176} a^{5} - \frac{15}{88} a^{4} + \frac{21}{176} a^{3} + \frac{7}{88} a^{2} - \frac{21}{44} a - \frac{1}{2}$, $\frac{1}{3520} a^{10} - \frac{9}{3520} a^{9} + \frac{191}{3520} a^{8} + \frac{107}{3520} a^{7} - \frac{83}{3520} a^{6} - \frac{243}{3520} a^{5} + \frac{125}{704} a^{4} + \frac{381}{3520} a^{3} + \frac{469}{1760} a^{2} - \frac{147}{880} a + \frac{17}{40}$, $\frac{1}{49280} a^{11} + \frac{3}{4928} a^{9} - \frac{1087}{24640} a^{8} - \frac{23}{1232} a^{7} - \frac{135}{4928} a^{6} - \frac{821}{24640} a^{5} - \frac{1957}{24640} a^{4} - \frac{559}{7040} a^{3} + \frac{8427}{24640} a^{2} + \frac{2491}{12320} a - \frac{21}{80}$, $\frac{1}{98560} a^{12} - \frac{1}{98560} a^{11} + \frac{1}{49280} a^{10} - \frac{17}{6160} a^{9} - \frac{87}{49280} a^{8} + \frac{247}{49280} a^{7} + \frac{1}{280} a^{6} - \frac{1527}{24640} a^{5} - \frac{12459}{98560} a^{4} + \frac{8419}{98560} a^{3} + \frac{16743}{49280} a^{2} - \frac{12249}{24640} a + \frac{73}{160}$, $\frac{1}{788480} a^{13} - \frac{1}{197120} a^{12} + \frac{3}{788480} a^{11} + \frac{3}{78848} a^{10} + \frac{173}{78848} a^{9} + \frac{431}{24640} a^{8} + \frac{7033}{394240} a^{7} - \frac{3867}{197120} a^{6} + \frac{1403}{22528} a^{5} + \frac{1241}{17920} a^{4} + \frac{18603}{788480} a^{3} - \frac{1677}{5120} a^{2} + \frac{63251}{197120} a + \frac{171}{1280}$, $\frac{1}{1576960} a^{14} - \frac{1}{1576960} a^{13} + \frac{1}{225280} a^{12} - \frac{9}{1576960} a^{11} + \frac{3}{78848} a^{10} + \frac{1459}{788480} a^{9} + \frac{587}{71680} a^{8} + \frac{673}{157696} a^{7} - \frac{68531}{1576960} a^{6} - \frac{26049}{1576960} a^{5} + \frac{188959}{1576960} a^{4} + \frac{387199}{1576960} a^{3} - \frac{15507}{112640} a^{2} - \frac{2159}{7168} a - \frac{719}{2560}$, $\frac{1}{84651212800} a^{15} + \frac{739}{42325606400} a^{14} + \frac{9931}{42325606400} a^{13} - \frac{1399}{755814400} a^{12} + \frac{330111}{84651212800} a^{11} - \frac{5315857}{42325606400} a^{10} - \frac{14416459}{10581401600} a^{9} - \frac{359676731}{10581401600} a^{8} + \frac{2258983519}{84651212800} a^{7} - \frac{2613678071}{42325606400} a^{6} - \frac{3025855801}{42325606400} a^{5} + \frac{28990561}{1322675200} a^{4} - \frac{15907360351}{84651212800} a^{3} + \frac{1893341939}{6046515200} a^{2} - \frac{76243929}{1923891200} a - \frac{6091269}{12492800}$, $\frac{1}{406720083429665525790058814453855867223262322704243025007411200} a^{16} + \frac{33392038030133845260381851367314436016149682824021}{12710002607177047680939337951682995850726947584507594531481600} a^{15} - \frac{36477336622326886653686352144480236330035941055163963743}{203360041714832762895029407226927933611631161352121512503705600} a^{14} + \frac{563559451914380689042594845801452082357685727460302667}{20336004171483276289502940722692793361163116135212151250370560} a^{13} + \frac{1874008874455671232485426537202853541988738742681594809679}{406720083429665525790058814453855867223262322704243025007411200} a^{12} - \frac{2392763444944714536843493318794374654648759091810532019}{264103950279003588175362866528477835859261248509248717537280} a^{11} + \frac{460683893711819991357956149102004436234779559368292257499}{14525717265345197349644957659066280972259368668008679464550400} a^{10} - \frac{87455615747976290136414412035390513088115461894074911062077}{50840010428708190723757351806731983402907790338030378125926400} a^{9} + \frac{1881355085711661130759792767474023041796254782332031582320527}{406720083429665525790058814453855867223262322704243025007411200} a^{8} - \frac{271899289213455401768588896013454145629125550748703315336411}{7262858632672598674822478829533140486129684334004339732275200} a^{7} + \frac{2374699513619054028761788798294254329220313583824310747915973}{40672008342966552579005881445385586722326232270424302500741120} a^{6} + \frac{181983149161660161590255078183546458364378728110623532952549}{9243638259765125586137700328496724255074143697823705113804800} a^{5} - \frac{15094891353734983040241175378907763599086089133297230475663203}{81344016685933105158011762890771173444652464540848605001482240} a^{4} + \frac{2012105863943536380398438532411937541746472094365417176395183}{9243638259765125586137700328496724255074143697823705113804800} a^{3} - \frac{312579925657868149228521886772686128500787815549820237758189}{50840010428708190723757351806731983402907790338030378125926400} a^{2} - \frac{295909451344445434198308183212305693189526564165024112719183}{2310909564941281396534425082124181063768535924455926278451200} a + \frac{12413047204522028326007985124443204936994613557768876964887}{30011812531704953201745780287327026802188778239687354265600}$

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:  $8$
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:  \( 5054960379310000000 \) (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 R $17$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{17}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ R $17$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $17$ $17$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $17$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
17Data not computed