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Properties

Label 17.17.236...361.1
Degree $17$
Signature $[17, 0]$
Discriminant $2.368\times 10^{39}$
Root discriminant $207.08$
Ramified prime $17$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{17}$ (as 17T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 136*x^15 - 85*x^14 + 6154*x^13 + 6545*x^12 - 119680*x^11 - 168555*x^10 + 998835*x^9 + 1749300*x^8 - 2783546*x^7 - 6581040*x^6 - 678725*x^5 + 3813882*x^4 + 770593*x^3 - 616267*x^2 - 82620*x + 577)
 
gp: K = bnfinit(x^17 - 136*x^15 - 85*x^14 + 6154*x^13 + 6545*x^12 - 119680*x^11 - 168555*x^10 + 998835*x^9 + 1749300*x^8 - 2783546*x^7 - 6581040*x^6 - 678725*x^5 + 3813882*x^4 + 770593*x^3 - 616267*x^2 - 82620*x + 577, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![577, -82620, -616267, 770593, 3813882, -678725, -6581040, -2783546, 1749300, 998835, -168555, -119680, 6545, 6154, -85, -136, 0, 1]);
 

\(x^{17} - 136 x^{15} - 85 x^{14} + 6154 x^{13} + 6545 x^{12} - 119680 x^{11} - 168555 x^{10} + 998835 x^{9} + 1749300 x^{8} - 2783546 x^{7} - 6581040 x^{6} - 678725 x^{5} + 3813882 x^{4} + 770593 x^{3} - 616267 x^{2} - 82620 x + 577\)  Toggle raw display

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

Invariants

Degree:  $17$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[17, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(2367911594760467245844106297320951247361\)\(\medspace = 17^{32}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $207.08$
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:  $17$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $17$
This field is Galois and abelian over $\Q$.
Conductor:  \(289=17^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{289}(256,·)$, $\chi_{289}(1,·)$, $\chi_{289}(69,·)$, $\chi_{289}(137,·)$, $\chi_{289}(205,·)$, $\chi_{289}(273,·)$, $\chi_{289}(18,·)$, $\chi_{289}(86,·)$, $\chi_{289}(154,·)$, $\chi_{289}(222,·)$, $\chi_{289}(35,·)$, $\chi_{289}(103,·)$, $\chi_{289}(171,·)$, $\chi_{289}(239,·)$, $\chi_{289}(52,·)$, $\chi_{289}(120,·)$, $\chi_{289}(188,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{131} a^{15} - \frac{18}{131} a^{14} + \frac{41}{131} a^{13} - \frac{11}{131} a^{12} + \frac{63}{131} a^{11} - \frac{46}{131} a^{10} + \frac{5}{131} a^{9} + \frac{33}{131} a^{8} - \frac{59}{131} a^{7} - \frac{64}{131} a^{6} - \frac{58}{131} a^{5} - \frac{21}{131} a^{4} - \frac{18}{131} a^{3} - \frac{47}{131} a^{2} + \frac{6}{131} a - \frac{54}{131}$, $\frac{1}{3944510245577144597573304311579244960447078599323} a^{16} - \frac{6174058074807975389663144027530413147904313786}{3944510245577144597573304311579244960447078599323} a^{15} - \frac{261522124122268955683359623478616415556821878084}{3944510245577144597573304311579244960447078599323} a^{14} - \frac{52976388106759767081722671050428240810850950117}{3944510245577144597573304311579244960447078599323} a^{13} - \frac{711713782268902153825598344991757002400564423614}{3944510245577144597573304311579244960447078599323} a^{12} - \frac{1212048790209943317515501798704916435380720019070}{3944510245577144597573304311579244960447078599323} a^{11} + \frac{345315770758007957297426072310132600566514628056}{3944510245577144597573304311579244960447078599323} a^{10} + \frac{436596642650347302239470754275084543915199424877}{3944510245577144597573304311579244960447078599323} a^{9} + \frac{1761965365504897130143282200300135802778129331328}{3944510245577144597573304311579244960447078599323} a^{8} - \frac{1720616893637595380571455992008401866720217677651}{3944510245577144597573304311579244960447078599323} a^{7} - \frac{1298618618900120233533833387313805841741365299492}{3944510245577144597573304311579244960447078599323} a^{6} - \frac{1129585891690935064018745119987947602182542299944}{3944510245577144597573304311579244960447078599323} a^{5} + \frac{505136008207589463224843019557726633812468141044}{3944510245577144597573304311579244960447078599323} a^{4} - \frac{1158174328533015240971064874702269170486415953767}{3944510245577144597573304311579244960447078599323} a^{3} + \frac{1215272207492939235485422888246272577657198676464}{3944510245577144597573304311579244960447078599323} a^{2} - \frac{133713698375855073379328585979803391705871929748}{3944510245577144597573304311579244960447078599323} a + \frac{682167394224991075241607018100575427678232941397}{3944510245577144597573304311579244960447078599323}$  Toggle raw display

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

Class group and class number

Trivial group, which has order $1$ (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:  $16$
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:  \( 111156254553000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{17}\cdot(2\pi)^{0}\cdot 111156254553000 \cdot 1}{2\sqrt{2367911594760467245844106297320951247361}}\approx 0.149703203263054$ (assuming GRH)

Galois group

$C_{17}$ (as 17T1):

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

Intermediate fields

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

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $17$ $17$ $17$ $17$ $17$ $17$ R $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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
17Data not computed