Properties

Label 17.3.275...423.1
Degree $17$
Signature $[3, 7]$
Discriminant $-2.757\times 10^{18}$
Root discriminant \(12.15\)
Ramified primes $75653,3119681,11682611$
Class number $1$
Class group trivial
Galois group $S_{17}$ (as 17T10)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 3*x^16 + 7*x^15 - 10*x^14 + 12*x^13 - 11*x^12 + 10*x^11 - 7*x^10 + 2*x^9 + 5*x^8 - 13*x^7 + 11*x^6 - 5*x^5 - 6*x^4 + 11*x^3 - 9*x^2 + 5*x - 1)
 
gp: K = bnfinit(y^17 - 3*y^16 + 7*y^15 - 10*y^14 + 12*y^13 - 11*y^12 + 10*y^11 - 7*y^10 + 2*y^9 + 5*y^8 - 13*y^7 + 11*y^6 - 5*y^5 - 6*y^4 + 11*y^3 - 9*y^2 + 5*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - 3*x^16 + 7*x^15 - 10*x^14 + 12*x^13 - 11*x^12 + 10*x^11 - 7*x^10 + 2*x^9 + 5*x^8 - 13*x^7 + 11*x^6 - 5*x^5 - 6*x^4 + 11*x^3 - 9*x^2 + 5*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - 3*x^16 + 7*x^15 - 10*x^14 + 12*x^13 - 11*x^12 + 10*x^11 - 7*x^10 + 2*x^9 + 5*x^8 - 13*x^7 + 11*x^6 - 5*x^5 - 6*x^4 + 11*x^3 - 9*x^2 + 5*x - 1)
 

\( x^{17} - 3 x^{16} + 7 x^{15} - 10 x^{14} + 12 x^{13} - 11 x^{12} + 10 x^{11} - 7 x^{10} + 2 x^{9} + 5 x^{8} - 13 x^{7} + 11 x^{6} - 5 x^{5} - 6 x^{4} + 11 x^{3} - 9 x^{2} + 5 x - 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $17$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[3, 7]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-2757250718309135423\) \(\medspace = -\,75653\cdot 3119681\cdot 11682611\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(12.15\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $75653^{1/2}3119681^{1/2}11682611^{1/2}\approx 1660497129.8707912$
Ramified primes:   \(75653\), \(3119681\), \(11682611\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  $\Q(\sqrt{-27572\!\cdots\!35423}$)
$\card{ \Aut(K/\Q) }$:  $1$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{109}a^{16}+\frac{6}{109}a^{15}-\frac{48}{109}a^{14}-\frac{6}{109}a^{13}-\frac{42}{109}a^{12}+\frac{47}{109}a^{11}-\frac{3}{109}a^{10}-\frac{34}{109}a^{9}+\frac{23}{109}a^{8}-\frac{6}{109}a^{7}+\frac{42}{109}a^{6}-\frac{47}{109}a^{5}+\frac{8}{109}a^{4}-\frac{43}{109}a^{3}-\frac{49}{109}a^{2}-\frac{14}{109}a-\frac{12}{109}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $a$, $\frac{49}{109}a^{16}-\frac{251}{109}a^{15}+\frac{482}{109}a^{14}-\frac{730}{109}a^{13}+\frac{667}{109}a^{12}-\frac{640}{109}a^{11}+\frac{507}{109}a^{10}-\frac{467}{109}a^{9}-\frac{72}{109}a^{8}+\frac{578}{109}a^{7}-\frac{885}{109}a^{6}+\frac{858}{109}a^{5}+\frac{174}{109}a^{4}-\frac{472}{109}a^{3}+\frac{760}{109}a^{2}-\frac{468}{109}a+\frac{175}{109}$, $\frac{175}{109}a^{16}-\frac{476}{109}a^{15}+\frac{865}{109}a^{14}-\frac{941}{109}a^{13}+\frac{716}{109}a^{12}-\frac{495}{109}a^{11}+\frac{456}{109}a^{10}-\frac{282}{109}a^{9}-\frac{553}{109}a^{8}+\frac{1130}{109}a^{7}-\frac{1479}{109}a^{6}+\frac{168}{109}a^{5}+\frac{1182}{109}a^{4}-\frac{1203}{109}a^{3}+\frac{799}{109}a^{2}+\frac{166}{109}a-\frac{247}{109}$, $\frac{22}{109}a^{16}-\frac{413}{109}a^{15}+\frac{1124}{109}a^{14}-\frac{2094}{109}a^{13}+\frac{2455}{109}a^{12}-\frac{2236}{109}a^{11}+\frac{1787}{109}a^{10}-\frac{1620}{109}a^{9}+\frac{942}{109}a^{8}+\frac{849}{109}a^{7}-\frac{2455}{109}a^{6}+\frac{3435}{109}a^{5}-\frac{1241}{109}a^{4}-\frac{1491}{109}a^{3}+\frac{2519}{109}a^{2}-\frac{2270}{109}a+\frac{608}{109}$, $\frac{246}{109}a^{16}-\frac{268}{109}a^{15}+\frac{400}{109}a^{14}+\frac{268}{109}a^{13}-\frac{413}{109}a^{12}+\frac{662}{109}a^{11}-\frac{193}{109}a^{10}+\frac{683}{109}a^{9}-\frac{991}{109}a^{8}+\frac{595}{109}a^{7}-\frac{241}{109}a^{6}-\frac{2297}{109}a^{5}+\frac{1205}{109}a^{4}-\frac{550}{109}a^{3}-\frac{827}{109}a^{2}+\frac{1243}{109}a-\frac{227}{109}$, $\frac{279}{109}a^{16}-\frac{506}{109}a^{15}+\frac{887}{109}a^{14}-\frac{584}{109}a^{13}+\frac{381}{109}a^{12}-\frac{76}{109}a^{11}+\frac{253}{109}a^{10}+\frac{215}{109}a^{9}-\frac{995}{109}a^{8}+\frac{1160}{109}a^{7}-\frac{1362}{109}a^{6}-\frac{1014}{109}a^{5}+\frac{1360}{109}a^{4}-\frac{1097}{109}a^{3}+\frac{390}{109}a^{2}+\frac{672}{109}a-\frac{187}{109}$, $\frac{106}{109}a^{16}-\frac{127}{109}a^{15}+\frac{144}{109}a^{14}+\frac{127}{109}a^{13}-\frac{310}{109}a^{12}+\frac{404}{109}a^{11}-\frac{318}{109}a^{10}+\frac{429}{109}a^{9}-\frac{614}{109}a^{8}+\frac{345}{109}a^{7}-\frac{235}{109}a^{6}-\frac{949}{109}a^{5}+\frac{848}{109}a^{4}-\frac{198}{109}a^{3}-\frac{180}{109}a^{2}+\frac{696}{109}a-\frac{291}{109}$, $\frac{155}{109}a^{16}-\frac{378}{109}a^{15}+\frac{626}{109}a^{14}-\frac{603}{109}a^{13}+\frac{357}{109}a^{12}-\frac{236}{109}a^{11}+\frac{189}{109}a^{10}-\frac{38}{109}a^{9}-\frac{686}{109}a^{8}+\frac{923}{109}a^{7}-\frac{1120}{109}a^{6}-\frac{91}{109}a^{5}+\frac{1022}{109}a^{4}-\frac{670}{109}a^{3}+\frac{471}{109}a^{2}+\frac{337}{109}a-\frac{225}{109}$, $\frac{211}{109}a^{16}-\frac{151}{109}a^{15}+\frac{118}{109}a^{14}+\frac{587}{109}a^{13}-\frac{687}{109}a^{12}+\frac{761}{109}a^{11}-\frac{306}{109}a^{10}+\frac{783}{109}a^{9}-\frac{924}{109}a^{8}+\frac{151}{109}a^{7}+\frac{360}{109}a^{6}-\frac{2396}{109}a^{5}+\frac{925}{109}a^{4}+\frac{192}{109}a^{3}-\frac{1074}{109}a^{2}+\frac{1188}{109}a-\frac{134}{109}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 178.334436956 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{7}\cdot 178.334436956 \cdot 1}{2\cdot\sqrt{2757250718309135423}}\cr\approx \mathstrut & 0.166079548501 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^17 - 3*x^16 + 7*x^15 - 10*x^14 + 12*x^13 - 11*x^12 + 10*x^11 - 7*x^10 + 2*x^9 + 5*x^8 - 13*x^7 + 11*x^6 - 5*x^5 - 6*x^4 + 11*x^3 - 9*x^2 + 5*x - 1)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^17 - 3*x^16 + 7*x^15 - 10*x^14 + 12*x^13 - 11*x^12 + 10*x^11 - 7*x^10 + 2*x^9 + 5*x^8 - 13*x^7 + 11*x^6 - 5*x^5 - 6*x^4 + 11*x^3 - 9*x^2 + 5*x - 1, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^17 - 3*x^16 + 7*x^15 - 10*x^14 + 12*x^13 - 11*x^12 + 10*x^11 - 7*x^10 + 2*x^9 + 5*x^8 - 13*x^7 + 11*x^6 - 5*x^5 - 6*x^4 + 11*x^3 - 9*x^2 + 5*x - 1);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - 3*x^16 + 7*x^15 - 10*x^14 + 12*x^13 - 11*x^12 + 10*x^11 - 7*x^10 + 2*x^9 + 5*x^8 - 13*x^7 + 11*x^6 - 5*x^5 - 6*x^4 + 11*x^3 - 9*x^2 + 5*x - 1);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$S_{17}$ (as 17T10):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A non-solvable group of order 355687428096000
The 297 conjugacy class representatives for $S_{17}$ are not computed
Character table for $S_{17}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $17$ ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ ${\href{/padicField/5.13.0.1}{13} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.7.0.1}{7} }$ ${\href{/padicField/11.13.0.1}{13} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ ${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ ${\href{/padicField/17.11.0.1}{11} }{,}\,{\href{/padicField/17.6.0.1}{6} }$ ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ $15{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.11.0.1}{11} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.6.0.1}{6} }$ $17$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(75653\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $7$$1$$7$$0$$C_7$$[\ ]^{7}$
\(3119681\) Copy content Toggle raw display $\Q_{3119681}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $12$$1$$12$$0$$C_{12}$$[\ ]^{12}$
\(11682611\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $15$$1$$15$$0$$C_{15}$$[\ ]^{15}$