Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^17 + 4*x^9 - 18*x^6 + 12*x^3 + 4*x - 2)
gp: K = bnfinit(y^17 + 4*y^9 - 18*y^6 + 12*y^3 + 4*y - 2, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 + 4*x^9 - 18*x^6 + 12*x^3 + 4*x - 2);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 + 4*x^9 - 18*x^6 + 12*x^3 + 4*x - 2)
\( x^{17} + 4x^{9} - 18x^{6} + 12x^{3} + 4x - 2 \)
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: |
\(63885424359899999574753280\)
\(\medspace = 2^{16}\cdot 5\cdot 7\cdot 27851834699314662203\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(32.96\) | 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: | $2^{16/17}5^{1/2}7^{1/2}27851834699314662203^{1/2}\approx 59949183925.2028$ | ||
| Ramified primes: |
\(2\), \(5\), \(7\), \(27851834699314662203\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{97481\!\cdots\!77105}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{367423}a^{16}-\frac{156495}{367423}a^{15}+\frac{104960}{367423}a^{14}-\frac{69985}{367423}a^{13}+\frac{157791}{367423}a^{12}-\frac{104984}{367423}a^{11}+\frac{151635}{367423}a^{10}-\frac{104870}{367423}a^{9}-\frac{52487}{367423}a^{8}-\frac{155523}{367423}a^{7}+\frac{104942}{367423}a^{6}+\frac{174946}{367423}a^{5}-\frac{16848}{367423}a^{4}+\frac{312}{367423}a^{3}+\frac{5833}{52489}a^{2}+\frac{864}{52489}a-\frac{108}{367423}$
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 $1$ (assuming GRH)
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: |
$a-1$, $\frac{57592}{367423}a^{16}+\frac{26150}{367423}a^{15}+\frac{13124}{367423}a^{14}+\frac{54190}{367423}a^{13}+\frac{26213}{367423}a^{12}+\frac{74360}{367423}a^{11}+\frac{53056}{367423}a^{10}+\frac{26234}{367423}a^{9}+\frac{325140}{367423}a^{8}+\frac{157278}{367423}a^{7}+\frac{78737}{367423}a^{6}-\frac{718320}{367423}a^{5}-\frac{313296}{367423}a^{4}-\frac{35023}{367423}a^{3}+\frac{4536}{52489}a^{2}-\frac{84}{52489}a+\frac{26255}{367423}$, $\frac{152392}{367423}a^{16}+\frac{106044}{367423}a^{15}+\frac{38861}{367423}a^{14}+\frac{33301}{367423}a^{13}+\frac{87837}{367423}a^{12}-\frac{22039}{367423}a^{11}-\frac{6396}{367423}a^{10}+\frac{81768}{367423}a^{9}+\frac{567229}{367423}a^{8}+\frac{527022}{367423}a^{7}+\frac{235189}{367423}a^{6}-\frac{2781432}{367423}a^{5}-\frac{1418184}{367423}a^{4}-\frac{586109}{367423}a^{3}+\frac{211277}{52489}a^{2}+\frac{76765}{52489}a+\frac{810545}{367423}$, $\frac{8748}{52489}a^{16}-\frac{162}{52489}a^{15}+\frac{3}{52489}a^{14}+\frac{2916}{52489}a^{13}-\frac{54}{52489}a^{12}+\frac{1}{52489}a^{11}+\frac{972}{52489}a^{10}-\frac{18}{52489}a^{9}+\frac{17496}{52489}a^{8}-\frac{324}{52489}a^{7}+\frac{6}{52489}a^{6}-\frac{151632}{52489}a^{5}-\frac{49681}{52489}a^{4}-\frac{52}{52489}a^{3}+\frac{106921}{52489}a^{2}+\frac{51481}{52489}a+\frac{52507}{52489}$, $\frac{14904}{52489}a^{16}-\frac{276}{52489}a^{15}-\frac{5827}{52489}a^{14}+\frac{4968}{52489}a^{13}-\frac{92}{52489}a^{12}+\frac{15554}{52489}a^{11}+\frac{1656}{52489}a^{10}-\frac{17527}{52489}a^{9}+\frac{82297}{52489}a^{8}-\frac{552}{52489}a^{7}-\frac{11654}{52489}a^{6}-\frac{205847}{52489}a^{5}-\frac{47705}{52489}a^{4}+\frac{135994}{52489}a^{3}+\frac{145225}{52489}a^{2}-\frac{36710}{52489}a-\frac{87451}{52489}$, $\frac{299088}{367423}a^{16}+\frac{239410}{367423}a^{15}+\frac{22783}{367423}a^{14}+\frac{47207}{367423}a^{13}+\frac{114796}{367423}a^{12}+\frac{147565}{367423}a^{11}+\frac{85721}{367423}a^{10}+\frac{73258}{367423}a^{9}+\frac{1385511}{367423}a^{8}+\frac{1056199}{367423}a^{7}+\frac{150544}{367423}a^{6}-\frac{4869258}{367423}a^{5}-\frac{3869832}{367423}a^{4}+\frac{357437}{367423}a^{3}+\frac{475812}{52489}a^{2}+\frac{166152}{52489}a-\frac{335703}{367423}$, $\frac{148685}{367423}a^{16}+\frac{72092}{367423}a^{15}+\frac{53098}{367423}a^{14}+\frac{67058}{367423}a^{13}+\frac{94016}{367423}a^{12}+\frac{52692}{367423}a^{11}+\frac{39849}{367423}a^{10}+\frac{101324}{367423}a^{9}+\frac{769771}{367423}a^{8}+\frac{196673}{367423}a^{7}+\frac{316152}{367423}a^{6}-\frac{2069813}{367423}a^{5}-\frac{1057135}{367423}a^{4}-\frac{640424}{367423}a^{3}+\frac{108836}{52489}a^{2}+\frac{23257}{52489}a+\frac{476055}{367423}$, $\frac{21780}{52489}a^{16}+\frac{17093}{52489}a^{15}+\frac{27872}{52489}a^{14}+\frac{7260}{52489}a^{13}+\frac{23194}{52489}a^{12}+\frac{26787}{52489}a^{11}+\frac{2420}{52489}a^{10}+\frac{42724}{52489}a^{9}+\frac{96049}{52489}a^{8}+\frac{86675}{52489}a^{7}+\frac{160722}{52489}a^{6}-\frac{377520}{52489}a^{5}-\frac{156308}{52489}a^{4}-\frac{395633}{52489}a^{3}+\frac{188009}{52489}a^{2}-\frac{74439}{52489}a+\frac{114743}{52489}$
| 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: | \( 2034427.1525 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 2034427.1525 \cdot 1}{2\cdot\sqrt{63885424359899999574753280}}\cr\approx \mathstrut & 0.61827288206 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^17 + 4*x^9 - 18*x^6 + 12*x^3 + 4*x - 2)
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^9 - 18*x^6 + 12*x^3 + 4*x - 2, 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^9 - 18*x^6 + 12*x^3 + 4*x - 2);
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^9 - 18*x^6 + 12*x^3 + 4*x - 2);
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 non-solvable group of order 355687428096000 |
| The 297 conjugacy class representatives for $S_{17}$ |
| Character table for $S_{17}$ |
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 | $17$ | R | R | ${\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.13.0.1}{13} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.8.0.1}{8} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.7.0.1}{7} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | $17$ | ${\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | $16{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.17.16.1 | $x^{17} + 2$ | $17$ | $1$ | $16$ | $C_{17}:C_{8}$ | $[\ ]_{17}^{8}$ |
|
\(5\)
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 5.12.0.1 | $x^{12} + x^{7} + x^{6} + 4 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.14.0.1 | $x^{14} + 5 x^{7} + 6 x^{5} + 2 x^{4} + 3 x^{2} + 6 x + 3$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
|
\(27851834699314662203\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |