Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 2*x^16 + 8*x^13 + 16*x^12 - 16*x^11 + 64*x^9 - 32*x^8 - 80*x^7 + 32*x^6 + 40*x^5 + 80*x^4 + 16*x^3 - 128*x^2 - 2*x + 68)
gp: K = bnfinit(y^17 - 2*y^16 + 8*y^13 + 16*y^12 - 16*y^11 + 64*y^9 - 32*y^8 - 80*y^7 + 32*y^6 + 40*y^5 + 80*y^4 + 16*y^3 - 128*y^2 - 2*y + 68, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - 2*x^16 + 8*x^13 + 16*x^12 - 16*x^11 + 64*x^9 - 32*x^8 - 80*x^7 + 32*x^6 + 40*x^5 + 80*x^4 + 16*x^3 - 128*x^2 - 2*x + 68);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - 2*x^16 + 8*x^13 + 16*x^12 - 16*x^11 + 64*x^9 - 32*x^8 - 80*x^7 + 32*x^6 + 40*x^5 + 80*x^4 + 16*x^3 - 128*x^2 - 2*x + 68)
\( x^{17} - 2 x^{16} + 8 x^{13} + 16 x^{12} - 16 x^{11} + 64 x^{9} - 32 x^{8} - 80 x^{7} + 32 x^{6} + 40 x^{5} + 80 x^{4} + 16 x^{3} - 128 x^{2} - 2 x + 68 \)
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: | $[1, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(604462909807314587353088\)
\(\medspace = 2^{79}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(25.06\) | 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: |
\(2\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{23\!\cdots\!82}a^{16}+\frac{76599019108737}{11\!\cdots\!91}a^{15}-\frac{31\!\cdots\!69}{11\!\cdots\!91}a^{14}-\frac{37\!\cdots\!82}{11\!\cdots\!91}a^{13}+\frac{54\!\cdots\!14}{11\!\cdots\!91}a^{12}+\frac{45\!\cdots\!51}{11\!\cdots\!91}a^{11}+\frac{44\!\cdots\!93}{11\!\cdots\!91}a^{10}+\frac{51\!\cdots\!16}{11\!\cdots\!91}a^{9}-\frac{29\!\cdots\!81}{11\!\cdots\!91}a^{8}-\frac{54\!\cdots\!87}{11\!\cdots\!91}a^{7}+\frac{17\!\cdots\!44}{11\!\cdots\!91}a^{6}-\frac{10\!\cdots\!65}{11\!\cdots\!91}a^{5}+\frac{33\!\cdots\!28}{11\!\cdots\!91}a^{4}-\frac{40\!\cdots\!18}{11\!\cdots\!91}a^{3}+\frac{57\!\cdots\!13}{11\!\cdots\!91}a^{2}-\frac{26\!\cdots\!51}{11\!\cdots\!91}a+\frac{24\!\cdots\!97}{11\!\cdots\!91}$
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: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(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: |
$\frac{10\!\cdots\!32}{11\!\cdots\!91}a^{16}-\frac{13\!\cdots\!38}{11\!\cdots\!91}a^{15}+\frac{153497539627694}{11\!\cdots\!91}a^{14}-\frac{20\!\cdots\!47}{11\!\cdots\!91}a^{13}+\frac{73\!\cdots\!74}{11\!\cdots\!91}a^{12}+\frac{19\!\cdots\!19}{11\!\cdots\!91}a^{11}+\frac{70\!\cdots\!54}{11\!\cdots\!91}a^{10}+\frac{19\!\cdots\!26}{11\!\cdots\!91}a^{9}+\frac{66\!\cdots\!91}{11\!\cdots\!91}a^{8}+\frac{15\!\cdots\!03}{11\!\cdots\!91}a^{7}-\frac{16\!\cdots\!82}{11\!\cdots\!91}a^{6}-\frac{10\!\cdots\!83}{11\!\cdots\!91}a^{5}-\frac{35\!\cdots\!48}{11\!\cdots\!91}a^{4}+\frac{42\!\cdots\!32}{11\!\cdots\!91}a^{3}+\frac{77\!\cdots\!97}{11\!\cdots\!91}a^{2}+\frac{35\!\cdots\!26}{11\!\cdots\!91}a-\frac{76\!\cdots\!89}{11\!\cdots\!91}$, $\frac{238227155182776}{11\!\cdots\!91}a^{16}-\frac{820973558914092}{11\!\cdots\!91}a^{15}+\frac{437201946462560}{11\!\cdots\!91}a^{14}+\frac{214894272296728}{11\!\cdots\!91}a^{13}+\frac{19\!\cdots\!89}{11\!\cdots\!91}a^{12}+\frac{16\!\cdots\!49}{11\!\cdots\!91}a^{11}-\frac{10\!\cdots\!58}{11\!\cdots\!91}a^{10}-\frac{68944875781624}{11\!\cdots\!91}a^{9}+\frac{95\!\cdots\!74}{11\!\cdots\!91}a^{8}-\frac{35\!\cdots\!73}{11\!\cdots\!91}a^{7}-\frac{25\!\cdots\!83}{11\!\cdots\!91}a^{6}+\frac{29\!\cdots\!46}{11\!\cdots\!91}a^{5}-\frac{21\!\cdots\!73}{11\!\cdots\!91}a^{4}-\frac{12\!\cdots\!16}{11\!\cdots\!91}a^{3}+\frac{86\!\cdots\!54}{11\!\cdots\!91}a^{2}-\frac{26\!\cdots\!39}{11\!\cdots\!91}a-\frac{19\!\cdots\!55}{11\!\cdots\!91}$, $\frac{361331936520056}{11\!\cdots\!91}a^{16}-\frac{395500407207728}{11\!\cdots\!91}a^{15}-\frac{404488405846192}{11\!\cdots\!91}a^{14}-\frac{477299961985816}{11\!\cdots\!91}a^{13}+\frac{31\!\cdots\!50}{11\!\cdots\!91}a^{12}+\frac{75\!\cdots\!71}{11\!\cdots\!91}a^{11}+\frac{19\!\cdots\!99}{11\!\cdots\!91}a^{10}-\frac{14\!\cdots\!16}{11\!\cdots\!91}a^{9}+\frac{22\!\cdots\!02}{11\!\cdots\!91}a^{8}+\frac{11\!\cdots\!03}{11\!\cdots\!91}a^{7}-\frac{33\!\cdots\!78}{11\!\cdots\!91}a^{6}-\frac{18\!\cdots\!13}{11\!\cdots\!91}a^{5}+\frac{10\!\cdots\!65}{11\!\cdots\!91}a^{4}+\frac{29\!\cdots\!40}{11\!\cdots\!91}a^{3}+\frac{21\!\cdots\!84}{11\!\cdots\!91}a^{2}-\frac{14\!\cdots\!61}{11\!\cdots\!91}a-\frac{10\!\cdots\!97}{11\!\cdots\!91}$, $\frac{541474688268459}{11\!\cdots\!91}a^{16}-\frac{503105829785995}{11\!\cdots\!91}a^{15}-\frac{759647397584390}{11\!\cdots\!91}a^{14}-\frac{700905168343220}{11\!\cdots\!91}a^{13}+\frac{43\!\cdots\!64}{11\!\cdots\!91}a^{12}+\frac{12\!\cdots\!98}{11\!\cdots\!91}a^{11}+\frac{40\!\cdots\!57}{11\!\cdots\!91}a^{10}-\frac{22\!\cdots\!00}{11\!\cdots\!91}a^{9}+\frac{31\!\cdots\!97}{11\!\cdots\!91}a^{8}+\frac{19\!\cdots\!09}{11\!\cdots\!91}a^{7}-\frac{42\!\cdots\!75}{11\!\cdots\!91}a^{6}-\frac{40\!\cdots\!01}{11\!\cdots\!91}a^{5}+\frac{57\!\cdots\!01}{11\!\cdots\!91}a^{4}+\frac{49\!\cdots\!83}{11\!\cdots\!91}a^{3}+\frac{49\!\cdots\!17}{11\!\cdots\!91}a^{2}-\frac{18\!\cdots\!89}{11\!\cdots\!91}a-\frac{37\!\cdots\!43}{11\!\cdots\!91}$, $\frac{49211265331281}{11\!\cdots\!91}a^{16}-\frac{455679016301548}{11\!\cdots\!91}a^{15}+\frac{604764560884272}{11\!\cdots\!91}a^{14}-\frac{220167784432132}{11\!\cdots\!91}a^{13}+\frac{11\!\cdots\!62}{11\!\cdots\!91}a^{12}-\frac{21\!\cdots\!68}{11\!\cdots\!91}a^{11}-\frac{66\!\cdots\!54}{11\!\cdots\!91}a^{10}+\frac{345221016246508}{11\!\cdots\!91}a^{9}-\frac{26\!\cdots\!13}{11\!\cdots\!91}a^{8}-\frac{23\!\cdots\!71}{11\!\cdots\!91}a^{7}-\frac{21\!\cdots\!05}{11\!\cdots\!91}a^{6}+\frac{12\!\cdots\!43}{11\!\cdots\!91}a^{5}+\frac{73\!\cdots\!29}{11\!\cdots\!91}a^{4}+\frac{10\!\cdots\!93}{11\!\cdots\!91}a^{3}-\frac{18\!\cdots\!40}{11\!\cdots\!91}a^{2}-\frac{20\!\cdots\!23}{11\!\cdots\!91}a+\frac{78\!\cdots\!63}{11\!\cdots\!91}$, $\frac{13\!\cdots\!49}{11\!\cdots\!91}a^{16}-\frac{19\!\cdots\!23}{11\!\cdots\!91}a^{15}-\frac{286237517021309}{11\!\cdots\!91}a^{14}-\frac{23\!\cdots\!39}{11\!\cdots\!91}a^{13}+\frac{10\!\cdots\!77}{11\!\cdots\!91}a^{12}+\frac{27\!\cdots\!63}{11\!\cdots\!91}a^{11}+\frac{14\!\cdots\!32}{11\!\cdots\!91}a^{10}+\frac{98\!\cdots\!34}{11\!\cdots\!91}a^{9}+\frac{75\!\cdots\!04}{11\!\cdots\!91}a^{8}+\frac{38\!\cdots\!35}{11\!\cdots\!91}a^{7}-\frac{72\!\cdots\!01}{11\!\cdots\!91}a^{6}-\frac{55\!\cdots\!00}{11\!\cdots\!91}a^{5}-\frac{28\!\cdots\!26}{11\!\cdots\!91}a^{4}+\frac{11\!\cdots\!10}{11\!\cdots\!91}a^{3}+\frac{90\!\cdots\!75}{11\!\cdots\!91}a^{2}-\frac{48\!\cdots\!36}{11\!\cdots\!91}a-\frac{39\!\cdots\!07}{11\!\cdots\!91}$, $\frac{133348952174370}{11\!\cdots\!91}a^{16}-\frac{621949317058588}{11\!\cdots\!91}a^{15}-\frac{12\!\cdots\!06}{11\!\cdots\!91}a^{14}+\frac{21\!\cdots\!88}{11\!\cdots\!91}a^{13}+\frac{14\!\cdots\!21}{11\!\cdots\!91}a^{12}+\frac{34\!\cdots\!20}{11\!\cdots\!91}a^{11}-\frac{22\!\cdots\!06}{11\!\cdots\!91}a^{10}-\frac{34\!\cdots\!84}{11\!\cdots\!91}a^{9}-\frac{10\!\cdots\!65}{11\!\cdots\!91}a^{8}-\frac{59\!\cdots\!82}{11\!\cdots\!91}a^{7}-\frac{12\!\cdots\!55}{11\!\cdots\!91}a^{6}-\frac{28\!\cdots\!18}{11\!\cdots\!91}a^{5}+\frac{35\!\cdots\!94}{11\!\cdots\!91}a^{4}+\frac{54\!\cdots\!04}{11\!\cdots\!91}a^{3}+\frac{23\!\cdots\!64}{11\!\cdots\!91}a^{2}-\frac{13\!\cdots\!15}{11\!\cdots\!91}a-\frac{12\!\cdots\!23}{11\!\cdots\!91}$, $\frac{636029286616479}{11\!\cdots\!91}a^{16}-\frac{20\!\cdots\!81}{11\!\cdots\!91}a^{15}+\frac{19\!\cdots\!57}{11\!\cdots\!91}a^{14}-\frac{466211932185150}{11\!\cdots\!91}a^{13}+\frac{43\!\cdots\!59}{11\!\cdots\!91}a^{12}+\frac{54\!\cdots\!40}{11\!\cdots\!91}a^{11}-\frac{21\!\cdots\!29}{11\!\cdots\!91}a^{10}+\frac{25\!\cdots\!16}{11\!\cdots\!91}a^{9}+\frac{37\!\cdots\!47}{11\!\cdots\!91}a^{8}-\frac{58\!\cdots\!17}{11\!\cdots\!91}a^{7}+\frac{15\!\cdots\!46}{11\!\cdots\!91}a^{6}+\frac{76\!\cdots\!77}{11\!\cdots\!91}a^{5}-\frac{11\!\cdots\!17}{11\!\cdots\!91}a^{4}+\frac{58\!\cdots\!54}{11\!\cdots\!91}a^{3}-\frac{83\!\cdots\!13}{11\!\cdots\!91}a^{2}-\frac{63\!\cdots\!59}{11\!\cdots\!91}a+\frac{13\!\cdots\!79}{11\!\cdots\!91}$
| 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: | \( 600164.932841 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 600164.932841 \cdot 1}{2\cdot\sqrt{604462909807314587353088}}\cr\approx \mathstrut & 1.87510129816 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^17 - 2*x^16 + 8*x^13 + 16*x^12 - 16*x^11 + 64*x^9 - 32*x^8 - 80*x^7 + 32*x^6 + 40*x^5 + 80*x^4 + 16*x^3 - 128*x^2 - 2*x + 68)
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 - 2*x^16 + 8*x^13 + 16*x^12 - 16*x^11 + 64*x^9 - 32*x^8 - 80*x^7 + 32*x^6 + 40*x^5 + 80*x^4 + 16*x^3 - 128*x^2 - 2*x + 68, 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 - 2*x^16 + 8*x^13 + 16*x^12 - 16*x^11 + 64*x^9 - 32*x^8 - 80*x^7 + 32*x^6 + 40*x^5 + 80*x^4 + 16*x^3 - 128*x^2 - 2*x + 68);
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 - 2*x^16 + 8*x^13 + 16*x^12 - 16*x^11 + 64*x^9 - 32*x^8 - 80*x^7 + 32*x^6 + 40*x^5 + 80*x^4 + 16*x^3 - 128*x^2 - 2*x + 68);
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
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 solvable group of order 272 |
| The 17 conjugacy class representatives for $F_{17}$ |
| Character table for $F_{17}$ |
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 | R | $16{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.8.0.1}{8} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.8.0.1}{8} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $17$ | $16{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.8.0.1}{8} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
In the table, R denotes a ramified prime. 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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 2.16.79.2 | $x^{16} + 16 x^{14} + 56 x^{12} + 48 x^{10} + 4 x^{8} + 32 x^{5} + 32 x^{4} + 32 x^{2} + 2$ | $16$ | $1$ | $79$ | $C_{16}$ | $[3, 4, 5, 6]$ |
Artin representations
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.
Additional information
This field is remarkable in that it is only ramified at 2, and
- it has the lowest degree of such a field where the degree is not a power of 2
- its Galois closure has the smallest degree for a Galois field where the degree is not a power of 2.
See Theorem 2.25 of [MR:1299733].