Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 51*x^15 + 1071*x^13 - 11934*x^11 + 75735*x^9 - 272646*x^7 + 520506*x^5 - 446148*x^3 + 111537*x - 22482)
gp: K = bnfinit(y^17 - 51*y^15 + 1071*y^13 - 11934*y^11 + 75735*y^9 - 272646*y^7 + 520506*y^5 - 446148*y^3 + 111537*y - 22482, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - 51*x^15 + 1071*x^13 - 11934*x^11 + 75735*x^9 - 272646*x^7 + 520506*x^5 - 446148*x^3 + 111537*x - 22482);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - 51*x^15 + 1071*x^13 - 11934*x^11 + 75735*x^9 - 272646*x^7 + 520506*x^5 - 446148*x^3 + 111537*x - 22482)
\( x^{17} - 51 x^{15} + 1071 x^{13} - 11934 x^{11} + 75735 x^{9} - 272646 x^{7} + 520506 x^{5} + \cdots - 22482 \)
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: | $[17, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(597436338855434422471226103296950272\)
\(\medspace = 2^{24}\cdot 3^{16}\cdot 17^{17}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(127.20\) | 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: | $2^{3/2}3^{16/17}17^{287/272}\approx 158.09016490784836$ | ||
| Ramified primes: |
\(2\), \(3\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{393}a^{9}-\frac{31}{131}a^{8}-\frac{9}{131}a^{7}-\frac{42}{131}a^{6}-\frac{50}{131}a^{5}+\frac{53}{131}a^{4}-\frac{8}{131}a^{3}+\frac{30}{131}a^{2}-\frac{19}{131}a-\frac{44}{131}$, $\frac{1}{393}a^{10}-\frac{10}{131}a^{8}+\frac{38}{131}a^{7}-\frac{26}{131}a^{6}-\frac{12}{131}a^{5}-\frac{57}{131}a^{4}-\frac{59}{131}a^{3}+\frac{20}{131}a^{2}+\frac{23}{131}a-\frac{31}{131}$, $\frac{1}{393}a^{11}+\frac{25}{131}a^{8}-\frac{34}{131}a^{7}+\frac{38}{131}a^{6}+\frac{15}{131}a^{5}-\frac{41}{131}a^{4}+\frac{42}{131}a^{3}+\frac{6}{131}a^{2}+\frac{54}{131}a-\frac{10}{131}$, $\frac{1}{393}a^{12}+\frac{64}{131}a^{8}+\frac{58}{131}a^{7}+\frac{21}{131}a^{6}+\frac{41}{131}a^{5}-\frac{3}{131}a^{4}-\frac{49}{131}a^{3}+\frac{31}{131}a^{2}-\frac{26}{131}a+\frac{25}{131}$, $\frac{1}{393}a^{13}-\frac{16}{131}a^{8}+\frac{46}{131}a^{7}-\frac{17}{131}a^{6}+\frac{34}{131}a^{5}-\frac{7}{131}a^{4}-\frac{5}{131}a^{3}-\frac{22}{131}a^{2}+\frac{5}{131}a+\frac{64}{131}$, $\frac{1}{393}a^{14}-\frac{1}{131}a^{8}-\frac{56}{131}a^{7}-\frac{17}{131}a^{6}-\frac{49}{131}a^{5}+\frac{50}{131}a^{4}-\frac{13}{131}a^{3}+\frac{4}{131}a^{2}-\frac{62}{131}a-\frac{16}{131}$, $\frac{1}{393}a^{15}-\frac{18}{131}a^{8}-\frac{44}{131}a^{7}-\frac{44}{131}a^{6}+\frac{31}{131}a^{5}+\frac{15}{131}a^{4}-\frac{20}{131}a^{3}+\frac{28}{131}a^{2}+\frac{58}{131}a-\frac{1}{131}$, $\frac{1}{393}a^{16}-\frac{15}{131}a^{8}-\frac{6}{131}a^{7}-\frac{10}{131}a^{6}-\frac{65}{131}a^{5}-\frac{40}{131}a^{4}-\frac{11}{131}a^{3}-\frac{25}{131}a^{2}+\frac{21}{131}a-\frac{18}{131}$
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: | $16$ | 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: |
$\frac{1}{393}a^{15}-\frac{1}{131}a^{14}-\frac{15}{131}a^{13}+\frac{122}{393}a^{12}+\frac{811}{393}a^{11}-\frac{634}{131}a^{10}-\frac{2486}{131}a^{9}+\frac{4692}{131}a^{8}+\frac{12226}{131}a^{7}-\frac{16334}{131}a^{6}-\frac{30062}{131}a^{5}+\frac{21744}{131}a^{4}+\frac{27814}{131}a^{3}-\frac{3488}{131}a^{2}-\frac{2228}{131}a-\frac{155}{131}$, $\frac{2}{393}a^{14}-\frac{77}{393}a^{12}+\frac{1144}{393}a^{10}-\frac{13}{393}a^{9}-\frac{2736}{131}a^{8}+\frac{104}{131}a^{7}+\frac{9630}{131}a^{6}-\frac{875}{131}a^{5}-\frac{14607}{131}a^{4}+\frac{3034}{131}a^{3}+\frac{5832}{131}a^{2}-\frac{2610}{131}a+\frac{355}{131}$, $\frac{1}{393}a^{16}+\frac{1}{393}a^{15}-\frac{50}{393}a^{14}-\frac{50}{393}a^{13}+\frac{340}{131}a^{12}+\frac{1016}{393}a^{11}-\frac{3630}{131}a^{10}-\frac{3571}{131}a^{9}+\frac{21600}{131}a^{8}+\frac{20622}{131}a^{7}-\frac{70364}{131}a^{6}-\frac{62811}{131}a^{5}+\frac{114649}{131}a^{4}+\frac{87473}{131}a^{3}-\frac{78358}{131}a^{2}-\frac{40147}{131}a+\frac{17543}{131}$, $\frac{1}{393}a^{15}-\frac{44}{393}a^{13}-\frac{7}{393}a^{12}+\frac{782}{393}a^{11}+\frac{84}{131}a^{10}-\frac{7160}{393}a^{9}-\frac{1157}{131}a^{8}+\frac{11811}{131}a^{7}+\frac{7552}{131}a^{6}-\frac{29749}{131}a^{5}-\frac{22999}{131}a^{4}+\frac{29949}{131}a^{3}+\frac{25747}{131}a^{2}-\frac{2984}{131}a-\frac{2395}{131}$, $\frac{1}{393}a^{15}-\frac{2}{393}a^{14}-\frac{17}{131}a^{13}+\frac{33}{131}a^{12}+\frac{348}{131}a^{11}-\frac{1939}{393}a^{10}-\frac{10922}{393}a^{9}+\frac{6330}{131}a^{8}+\frac{20487}{131}a^{7}-\frac{32121}{131}a^{6}-\frac{59920}{131}a^{5}+\frac{78590}{131}a^{4}+\frac{82394}{131}a^{3}-\frac{75494}{131}a^{2}-\frac{45612}{131}a+\frac{17269}{131}$, $\frac{2}{393}a^{16}+\frac{1}{131}a^{15}-\frac{98}{393}a^{14}-\frac{146}{393}a^{13}+\frac{1940}{393}a^{12}+\frac{2879}{393}a^{11}-\frac{6584}{131}a^{10}-\frac{29318}{393}a^{9}+\frac{36128}{131}a^{8}+\frac{53954}{131}a^{7}-\frac{100800}{131}a^{6}-\frac{152912}{131}a^{5}+\frac{114696}{131}a^{4}+\frac{180441}{131}a^{3}-\frac{22386}{131}a^{2}-\frac{41985}{131}a+\frac{8935}{131}$, $\frac{1}{393}a^{15}-\frac{2}{393}a^{14}-\frac{14}{131}a^{13}+\frac{80}{393}a^{12}+\frac{692}{393}a^{11}-\frac{1222}{393}a^{10}-\frac{5650}{393}a^{9}+\frac{2942}{131}a^{8}+\frac{7897}{131}a^{7}-\frac{9967}{131}a^{6}-\frac{15617}{131}a^{5}+\frac{12840}{131}a^{4}+\frac{9824}{131}a^{3}-\frac{1130}{131}a^{2}+\frac{2499}{131}a-\frac{911}{131}$, $\frac{2}{393}a^{16}-\frac{5}{393}a^{15}-\frac{31}{131}a^{14}+\frac{78}{131}a^{13}+\frac{1720}{393}a^{12}-\frac{1459}{131}a^{11}-\frac{15976}{393}a^{10}+\frac{13823}{131}a^{9}+\frac{25657}{131}a^{8}-\frac{69066}{131}a^{7}-\frac{58859}{131}a^{6}+\frac{170833}{131}a^{5}+\frac{49027}{131}a^{4}-\frac{173829}{131}a^{3}-\frac{5817}{131}a^{2}+\frac{53695}{131}a-\frac{11545}{131}$, $\frac{2}{393}a^{14}-\frac{4}{393}a^{13}-\frac{95}{393}a^{12}+\frac{157}{393}a^{11}+\frac{1805}{393}a^{10}-\frac{781}{131}a^{9}-\frac{5805}{131}a^{8}+\frac{5435}{131}a^{7}+\frac{29674}{131}a^{6}-\frac{16848}{131}a^{5}-\frac{75102}{131}a^{4}+\frac{14668}{131}a^{3}+\frac{73026}{131}a^{2}+\frac{11817}{131}a-\frac{9365}{131}$, $\frac{1}{393}a^{16}-\frac{1}{131}a^{15}-\frac{14}{131}a^{14}+\frac{41}{131}a^{13}+\frac{715}{393}a^{12}-\frac{673}{131}a^{11}-\frac{6350}{393}a^{10}+\frac{5656}{131}a^{9}+\frac{10481}{131}a^{8}-\frac{25812}{131}a^{7}-\frac{28354}{131}a^{6}+\frac{62404}{131}a^{5}+\frac{36702}{131}a^{4}-\frac{70897}{131}a^{3}-\frac{13586}{131}a^{2}+\frac{25721}{131}a-\frac{4955}{131}$, $\frac{1}{393}a^{16}+\frac{2}{393}a^{15}-\frac{43}{393}a^{14}-\frac{30}{131}a^{13}+\frac{719}{393}a^{12}+\frac{1598}{393}a^{11}-\frac{5830}{393}a^{10}-\frac{4708}{131}a^{9}+\frac{7666}{131}a^{8}+\frac{21465}{131}a^{7}-\frac{12089}{131}a^{6}-\frac{47344}{131}a^{5}-\frac{243}{131}a^{4}+\frac{41916}{131}a^{3}+\frac{12418}{131}a^{2}-\frac{5776}{131}a-\frac{11}{131}$, $\frac{1}{131}a^{16}+\frac{4}{393}a^{15}-\frac{133}{393}a^{14}-\frac{181}{393}a^{13}+\frac{2332}{393}a^{12}+\frac{1079}{131}a^{11}-\frac{20500}{393}a^{10}-\frac{28852}{393}a^{9}+\frac{31536}{131}a^{8}+\frac{44252}{131}a^{7}-\frac{73367}{131}a^{6}-\frac{98798}{131}a^{5}+\frac{75959}{131}a^{4}+\frac{90988}{131}a^{3}-\frac{23760}{131}a^{2}-\frac{18839}{131}a+\frac{5005}{131}$, $\frac{1}{393}a^{16}-\frac{2}{393}a^{15}-\frac{46}{393}a^{14}+\frac{26}{131}a^{13}+\frac{875}{393}a^{12}-\frac{1175}{393}a^{11}-\frac{2959}{131}a^{10}+\frac{8477}{393}a^{9}+\frac{17142}{131}a^{8}-\frac{9609}{131}a^{7}-\frac{55950}{131}a^{6}+\frac{11457}{131}a^{5}+\frac{94266}{131}a^{4}+\frac{5654}{131}a^{3}-\frac{68231}{131}a^{2}-\frac{14022}{131}a+\frac{11359}{131}$, $\frac{7}{393}a^{15}+\frac{3}{131}a^{14}-\frac{99}{131}a^{13}-\frac{329}{393}a^{12}+\frac{5005}{393}a^{11}+\frac{4532}{393}a^{10}-\frac{325}{3}a^{9}-\frac{9732}{131}a^{8}+\frac{63727}{131}a^{7}+\frac{29689}{131}a^{6}-\frac{142504}{131}a^{5}-\frac{39660}{131}a^{4}+\frac{131912}{131}a^{3}+\frac{22131}{131}a^{2}-\frac{32275}{131}a+\frac{4643}{131}$, $\frac{1}{393}a^{16}+\frac{7}{393}a^{15}-\frac{37}{393}a^{14}-\frac{289}{393}a^{13}+\frac{515}{393}a^{12}+\frac{4703}{393}a^{11}-\frac{1095}{131}a^{10}-\frac{38281}{393}a^{9}+\frac{2915}{131}a^{8}+\frac{54289}{131}a^{7}-\frac{269}{131}a^{6}-\frac{113882}{131}a^{5}-\frac{12090}{131}a^{4}+\frac{96812}{131}a^{3}+\frac{19233}{131}a^{2}-\frac{17735}{131}a+\frac{461}{131}$, $\frac{1}{393}a^{16}+\frac{4}{393}a^{15}-\frac{43}{393}a^{14}-\frac{178}{393}a^{13}+\frac{716}{393}a^{12}+\frac{3140}{393}a^{11}-\frac{5725}{393}a^{10}-\frac{9281}{131}a^{9}+\frac{7240}{131}a^{8}+\frac{43325}{131}a^{7}-\frac{10100}{131}a^{6}-\frac{102246}{131}a^{5}-\frac{2173}{131}a^{4}+\frac{107556}{131}a^{3}+\frac{7492}{131}a^{2}-\frac{32897}{131}a+\frac{6509}{131}$
| 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: | \( 13029132161200 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{17}\cdot(2\pi)^{0}\cdot 13029132161200 \cdot 1}{2\cdot\sqrt{597436338855434422471226103296950272}}\cr\approx \mathstrut & 1.10471335064194 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^17 - 51*x^15 + 1071*x^13 - 11934*x^11 + 75735*x^9 - 272646*x^7 + 520506*x^5 - 446148*x^3 + 111537*x - 22482)
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 - 51*x^15 + 1071*x^13 - 11934*x^11 + 75735*x^9 - 272646*x^7 + 520506*x^5 - 446148*x^3 + 111537*x - 22482, 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 - 51*x^15 + 1071*x^13 - 11934*x^11 + 75735*x^9 - 272646*x^7 + 520506*x^5 - 446148*x^3 + 111537*x - 22482);
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 - 51*x^15 + 1071*x^13 - 11934*x^11 + 75735*x^9 - 272646*x^7 + 520506*x^5 - 446148*x^3 + 111537*x - 22482);
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 | R | $16{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | R | ${\href{/padicField/19.8.0.1}{8} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.8.0.1}{8} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 | $[\ ]$ |
| 2.8.12.6 | $x^{8} - 16 x^{6} + 72 x^{4} + 3664$ | $2$ | $4$ | $12$ | $C_8$ | $[3]^{4}$ | |
| 2.8.12.6 | $x^{8} - 16 x^{6} + 72 x^{4} + 3664$ | $2$ | $4$ | $12$ | $C_8$ | $[3]^{4}$ | |
|
\(3\)
| 3.17.16.1 | $x^{17} + 3$ | $17$ | $1$ | $16$ | $F_{17}$ | $[\ ]_{17}^{16}$ |
|
\(17\)
| 17.17.17.1 | $x^{17} + 17 x + 17$ | $17$ | $1$ | $17$ | $F_{17}$ | $[17/16]_{16}$ |