Properties

 Label 17.3.377...359.1 Degree $17$ Signature $[3, 7]$ Discriminant $-3.780\times 10^{18}$ Root discriminant $$12.38$$ Ramified primes $137,1926157,14323744651$ Class number $1$ Class group trivial Galois group $S_{17}$ (as 17T10)

Related objects

Show commands: Magma / Oscar / PariGP / SageMath

Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 4*x^16 + 12*x^15 - 26*x^14 + 48*x^13 - 75*x^12 + 105*x^11 - 126*x^10 + 136*x^9 - 129*x^8 + 107*x^7 - 80*x^6 + 47*x^5 - 23*x^4 + 11*x^3 - 2*x^2 - 2*x + 1)

gp: K = bnfinit(y^17 - 4*y^16 + 12*y^15 - 26*y^14 + 48*y^13 - 75*y^12 + 105*y^11 - 126*y^10 + 136*y^9 - 129*y^8 + 107*y^7 - 80*y^6 + 47*y^5 - 23*y^4 + 11*y^3 - 2*y^2 - 2*y + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - 4*x^16 + 12*x^15 - 26*x^14 + 48*x^13 - 75*x^12 + 105*x^11 - 126*x^10 + 136*x^9 - 129*x^8 + 107*x^7 - 80*x^6 + 47*x^5 - 23*x^4 + 11*x^3 - 2*x^2 - 2*x + 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - 4*x^16 + 12*x^15 - 26*x^14 + 48*x^13 - 75*x^12 + 105*x^11 - 126*x^10 + 136*x^9 - 129*x^8 + 107*x^7 - 80*x^6 + 47*x^5 - 23*x^4 + 11*x^3 - 2*x^2 - 2*x + 1)

$$x^{17} - 4 x^{16} + 12 x^{15} - 26 x^{14} + 48 x^{13} - 75 x^{12} + 105 x^{11} - 126 x^{10} + 136 x^{9} - 129 x^{8} + 107 x^{7} - 80 x^{6} + 47 x^{5} - 23 x^{4} + 11 x^{3} - 2 x^{2} - 2 x + 1$$

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: $$-3779800000525860359$$ -3779800000525860359 $$\medspace = -\,137\cdot 1926157\cdot 14323744651$$ sage: K.disc()  gp: K.disc  magma: OK := Integers(K); Discriminant(OK);  oscar: OK = ring_of_integers(K); discriminant(OK) Root discriminant: $$12.38$$ 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)) Ramified primes: $$137$$, $$1926157$$, $$14323744651$$ 137, 1926157, 14323744651 sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(OK));  oscar: prime_divisors(discriminant((OK))) Discriminant root field: $\Q(\sqrt{-37798\!\cdots\!60359}$) $\card{ \Aut(K/\Q) }$: $1$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{11}-\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{7}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-\frac{1}{3}a^{8}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{12}-\frac{1}{3}a^{10}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{16}+\frac{1}{3}a^{11}-\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}$

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: $$-1$$ -1  (order $2$) 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: $\frac{8}{3}a^{16}-\frac{25}{3}a^{15}+\frac{74}{3}a^{14}-\frac{142}{3}a^{13}+\frac{254}{3}a^{12}-\frac{362}{3}a^{11}+\frac{491}{3}a^{10}-\frac{527}{3}a^{9}+\frac{557}{3}a^{8}-\frac{463}{3}a^{7}+122a^{6}-82a^{5}+\frac{104}{3}a^{4}-\frac{55}{3}a^{3}+\frac{14}{3}a^{2}+\frac{11}{3}a-3$, $\frac{4}{3}a^{16}-4a^{15}+\frac{35}{3}a^{14}-\frac{67}{3}a^{13}+\frac{121}{3}a^{12}-59a^{11}+\frac{245}{3}a^{10}-90a^{9}+97a^{8}-\frac{254}{3}a^{7}+\frac{203}{3}a^{6}-\frac{145}{3}a^{5}+\frac{61}{3}a^{4}-\frac{34}{3}a^{3}+6a^{2}+3a-2$, $2a^{16}-6a^{15}+\frac{52}{3}a^{14}-\frac{97}{3}a^{13}+\frac{173}{3}a^{12}-82a^{11}+\frac{338}{3}a^{10}-\frac{364}{3}a^{9}+\frac{392}{3}a^{8}-\frac{332}{3}a^{7}+90a^{6}-\frac{185}{3}a^{5}+\frac{79}{3}a^{4}-15a^{3}+\frac{14}{3}a^{2}+\frac{7}{3}a-\frac{8}{3}$, $a$, $\frac{2}{3}a^{16}-\frac{7}{3}a^{15}+\frac{20}{3}a^{14}-\frac{40}{3}a^{13}+\frac{71}{3}a^{12}-\frac{104}{3}a^{11}+\frac{140}{3}a^{10}-\frac{155}{3}a^{9}+\frac{161}{3}a^{8}-\frac{142}{3}a^{7}+38a^{6}-27a^{5}+\frac{47}{3}a^{4}-\frac{25}{3}a^{3}+\frac{17}{3}a^{2}-\frac{1}{3}a-1$, $\frac{4}{3}a^{16}-\frac{13}{3}a^{15}+\frac{38}{3}a^{14}-24a^{13}+\frac{125}{3}a^{12}-\frac{172}{3}a^{11}+\frac{224}{3}a^{10}-\frac{227}{3}a^{9}+75a^{8}-\frac{167}{3}a^{7}+\frac{118}{3}a^{6}-22a^{5}+\frac{11}{3}a^{4}-\frac{2}{3}a^{3}-\frac{5}{3}a^{2}+3a-2$, $a^{16}-3a^{15}+\frac{25}{3}a^{14}-\frac{46}{3}a^{13}+\frac{77}{3}a^{12}-35a^{11}+\frac{134}{3}a^{10}-\frac{136}{3}a^{9}+\frac{134}{3}a^{8}-\frac{110}{3}a^{7}+25a^{6}-\frac{62}{3}a^{5}+\frac{16}{3}a^{4}-6a^{3}+\frac{8}{3}a^{2}+\frac{4}{3}a+\frac{1}{3}$, $\frac{2}{3}a^{16}-3a^{15}+\frac{26}{3}a^{14}-19a^{13}+\frac{100}{3}a^{12}-51a^{11}+68a^{10}-\frac{241}{3}a^{9}+\frac{245}{3}a^{8}-77a^{7}+\frac{178}{3}a^{6}-\frac{136}{3}a^{5}+26a^{4}-\frac{32}{3}a^{3}+\frac{20}{3}a^{2}-\frac{5}{3}a-\frac{2}{3}$, $\frac{7}{3}a^{16}-8a^{15}+\frac{71}{3}a^{14}-\frac{145}{3}a^{13}+\frac{265}{3}a^{12}-134a^{11}+\frac{560}{3}a^{10}-216a^{9}+233a^{8}-\frac{641}{3}a^{7}+\frac{524}{3}a^{6}-\frac{385}{3}a^{5}+\frac{202}{3}a^{4}-\frac{103}{3}a^{3}+17a^{2}+a-4$ 8/3*a^16 - 25/3*a^15 + 74/3*a^14 - 142/3*a^13 + 254/3*a^12 - 362/3*a^11 + 491/3*a^10 - 527/3*a^9 + 557/3*a^8 - 463/3*a^7 + 122*a^6 - 82*a^5 + 104/3*a^4 - 55/3*a^3 + 14/3*a^2 + 11/3*a - 3, 4/3*a^16 - 4*a^15 + 35/3*a^14 - 67/3*a^13 + 121/3*a^12 - 59*a^11 + 245/3*a^10 - 90*a^9 + 97*a^8 - 254/3*a^7 + 203/3*a^6 - 145/3*a^5 + 61/3*a^4 - 34/3*a^3 + 6*a^2 + 3*a - 2, 2*a^16 - 6*a^15 + 52/3*a^14 - 97/3*a^13 + 173/3*a^12 - 82*a^11 + 338/3*a^10 - 364/3*a^9 + 392/3*a^8 - 332/3*a^7 + 90*a^6 - 185/3*a^5 + 79/3*a^4 - 15*a^3 + 14/3*a^2 + 7/3*a - 8/3, a, 2/3*a^16 - 7/3*a^15 + 20/3*a^14 - 40/3*a^13 + 71/3*a^12 - 104/3*a^11 + 140/3*a^10 - 155/3*a^9 + 161/3*a^8 - 142/3*a^7 + 38*a^6 - 27*a^5 + 47/3*a^4 - 25/3*a^3 + 17/3*a^2 - 1/3*a - 1, 4/3*a^16 - 13/3*a^15 + 38/3*a^14 - 24*a^13 + 125/3*a^12 - 172/3*a^11 + 224/3*a^10 - 227/3*a^9 + 75*a^8 - 167/3*a^7 + 118/3*a^6 - 22*a^5 + 11/3*a^4 - 2/3*a^3 - 5/3*a^2 + 3*a - 2, a^16 - 3*a^15 + 25/3*a^14 - 46/3*a^13 + 77/3*a^12 - 35*a^11 + 134/3*a^10 - 136/3*a^9 + 134/3*a^8 - 110/3*a^7 + 25*a^6 - 62/3*a^5 + 16/3*a^4 - 6*a^3 + 8/3*a^2 + 4/3*a + 1/3, 2/3*a^16 - 3*a^15 + 26/3*a^14 - 19*a^13 + 100/3*a^12 - 51*a^11 + 68*a^10 - 241/3*a^9 + 245/3*a^8 - 77*a^7 + 178/3*a^6 - 136/3*a^5 + 26*a^4 - 32/3*a^3 + 20/3*a^2 - 5/3*a - 2/3, 7/3*a^16 - 8*a^15 + 71/3*a^14 - 145/3*a^13 + 265/3*a^12 - 134*a^11 + 560/3*a^10 - 216*a^9 + 233*a^8 - 641/3*a^7 + 524/3*a^6 - 385/3*a^5 + 202/3*a^4 - 103/3*a^3 + 17*a^2 + a - 4 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: $$206.585713852$$ 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 206.585713852 \cdot 1}{2\cdot\sqrt{3779800000525860359}}\cr\approx \mathstrut & 0.164317925990 \end{aligned}

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^17 - 4*x^16 + 12*x^15 - 26*x^14 + 48*x^13 - 75*x^12 + 105*x^11 - 126*x^10 + 136*x^9 - 129*x^8 + 107*x^7 - 80*x^6 + 47*x^5 - 23*x^4 + 11*x^3 - 2*x^2 - 2*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 - 4*x^16 + 12*x^15 - 26*x^14 + 48*x^13 - 75*x^12 + 105*x^11 - 126*x^10 + 136*x^9 - 129*x^8 + 107*x^7 - 80*x^6 + 47*x^5 - 23*x^4 + 11*x^3 - 2*x^2 - 2*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 - 4*x^16 + 12*x^15 - 26*x^14 + 48*x^13 - 75*x^12 + 105*x^11 - 126*x^10 + 136*x^9 - 129*x^8 + 107*x^7 - 80*x^6 + 47*x^5 - 23*x^4 + 11*x^3 - 2*x^2 - 2*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 - 4*x^16 + 12*x^15 - 26*x^14 + 48*x^13 - 75*x^12 + 105*x^11 - 126*x^10 + 136*x^9 - 129*x^8 + 107*x^7 - 80*x^6 + 47*x^5 - 23*x^4 + 11*x^3 - 2*x^2 - 2*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 $17$ ${\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.4.0.1}{4} }^{2}$ $17$ $17$ ${\href{/padicField/11.13.0.1}{13} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ $17$ ${\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.8.0.1}{8} }$ ${\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.8.0.1}{8} }$ ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ $17$ ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.7.0.1}{7} }$ $17$

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]])