Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 - 27*x^12 + 44*x^10 + 304*x^8 + 1320*x^6 + 5093*x^4 - 14296*x^2 + 10201)
gp: K = bnfinit(y^16 - 4*y^14 - 27*y^12 + 44*y^10 + 304*y^8 + 1320*y^6 + 5093*y^4 - 14296*y^2 + 10201, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^14 - 27*x^12 + 44*x^10 + 304*x^8 + 1320*x^6 + 5093*x^4 - 14296*x^2 + 10201);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^14 - 27*x^12 + 44*x^10 + 304*x^8 + 1320*x^6 + 5093*x^4 - 14296*x^2 + 10201)
\( x^{16} - 4x^{14} - 27x^{12} + 44x^{10} + 304x^{8} + 1320x^{6} + 5093x^{4} - 14296x^{2} + 10201 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(723177258444187191407017984\)
\(\medspace = 2^{32}\cdot 17^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(47.72\) | 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: | $2^{2}17^{7/8}\approx 47.72025959461256$ | ||
| Ramified primes: |
\(2\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(136=2^{3}\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{136}(1,·)$, $\chi_{136}(67,·)$, $\chi_{136}(77,·)$, $\chi_{136}(15,·)$, $\chi_{136}(81,·)$, $\chi_{136}(87,·)$, $\chi_{136}(89,·)$, $\chi_{136}(93,·)$, $\chi_{136}(33,·)$, $\chi_{136}(35,·)$, $\chi_{136}(111,·)$, $\chi_{136}(115,·)$, $\chi_{136}(53,·)$, $\chi_{136}(123,·)$, $\chi_{136}(117,·)$, $\chi_{136}(127,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{4}a^{12}+\frac{1}{4}$, $\frac{1}{404}a^{13}+\frac{34}{101}a^{11}-\frac{19}{101}a^{9}-\frac{23}{101}a^{7}-\frac{13}{101}a^{5}+\frac{25}{101}a^{3}+\frac{105}{404}a$, $\frac{1}{106934489978116}a^{14}-\frac{2979271606241}{106934489978116}a^{12}-\frac{6546518867105}{26733622494529}a^{10}+\frac{8590981813554}{26733622494529}a^{8}+\frac{8370602232817}{26733622494529}a^{6}+\frac{5203225098104}{26733622494529}a^{4}+\frac{30764870783529}{106934489978116}a^{2}+\frac{450690051659}{1058757326516}$, $\frac{1}{106934489978116}a^{15}-\frac{33844479161}{53467244989058}a^{13}+\frac{12246423678554}{26733622494529}a^{11}+\frac{6738156492151}{26733622494529}a^{9}-\frac{5128553680262}{26733622494529}a^{7}-\frac{5913726830314}{26733622494529}a^{5}+\frac{1119665641081}{106934489978116}a^{3}+\frac{15216201657353}{53467244989058}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{2}\times C_{2}\times C_{8}$, which has order $32$ (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: | $7$ | 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: |
$\frac{4988903883}{106934489978116}a^{14}-\frac{12706893059}{53467244989058}a^{12}-\frac{17843641928}{26733622494529}a^{10}+\frac{75183729980}{26733622494529}a^{8}+\frac{39553438770}{26733622494529}a^{6}+\frac{873921225780}{26733622494529}a^{4}-\frac{7873762705105}{106934489978116}a^{2}-\frac{232564881219}{529378663258}$, $\frac{2628959753}{53467244989058}a^{14}-\frac{12156118611}{26733622494529}a^{12}-\frac{23481921112}{26733622494529}a^{10}+\frac{255675604093}{26733622494529}a^{8}+\frac{600369750292}{26733622494529}a^{6}+\frac{958351628208}{26733622494529}a^{4}-\frac{8153810538315}{53467244989058}a^{2}-\frac{737013518559}{264689331629}$, $\frac{4725930809}{106934489978116}a^{14}-\frac{6514856015}{53467244989058}a^{12}-\frac{41824735676}{26733622494529}a^{10}+\frac{100456869226}{26733622494529}a^{8}+\frac{472996840542}{26733622494529}a^{6}-\frac{1672439143227}{26733622494529}a^{4}+\frac{7401091865845}{106934489978116}a^{2}-\frac{90539987211}{529378663258}$, $\frac{3185582257}{26733622494529}a^{15}+\frac{5636803273}{26733622494529}a^{14}-\frac{28113617809}{53467244989058}a^{13}-\frac{35766258931}{106934489978116}a^{12}-\frac{73854602328}{26733622494529}a^{11}-\frac{230419180434}{26733622494529}a^{10}+\frac{163647540420}{26733622494529}a^{9}+\frac{39014207034}{26733622494529}a^{8}+\frac{712741402035}{26733622494529}a^{7}+\frac{2336958790386}{26733622494529}a^{6}+\frac{3604598828948}{26733622494529}a^{5}+\frac{12251550635904}{26733622494529}a^{4}+\frac{15265818806693}{26733622494529}a^{3}+\frac{34401431227231}{26733622494529}a^{2}-\frac{29461112364771}{53467244989058}a-\frac{1701741042867}{1058757326516}$, $\frac{50339198061}{106934489978116}a^{15}+\frac{29913710323}{53467244989058}a^{14}-\frac{131808494357}{106934489978116}a^{13}-\frac{153622422455}{106934489978116}a^{12}-\frac{390512720622}{26733622494529}a^{11}-\frac{491112451605}{26733622494529}a^{10}-\frac{26669023470}{26733622494529}a^{9}-\frac{4010626544}{26733622494529}a^{8}+\frac{4203174038657}{26733622494529}a^{7}+\frac{5986259428325}{26733622494529}a^{6}+\frac{24105004781061}{26733622494529}a^{5}+\frac{27626824252858}{26733622494529}a^{4}+\frac{353843041200001}{106934489978116}a^{3}+\frac{181642208449009}{53467244989058}a^{2}-\frac{349996433221873}{106934489978116}a-\frac{4356689396551}{1058757326516}$, $\frac{18788817426}{26733622494529}a^{15}-\frac{1041872937}{1137600957214}a^{14}-\frac{108221924723}{53467244989058}a^{13}+\frac{6359734331}{2275201914428}a^{12}-\frac{569151540309}{26733622494529}a^{11}+\frac{15657761348}{568800478607}a^{10}+\frac{207229531032}{26733622494529}a^{9}-\frac{6522823698}{568800478607}a^{8}+\frac{6037690577484}{26733622494529}a^{7}-\frac{169441717823}{568800478607}a^{6}+\frac{31463383421835}{26733622494529}a^{5}-\frac{857767633818}{568800478607}a^{4}+\frac{129872937309032}{26733622494529}a^{3}-\frac{7023296903861}{1137600957214}a^{2}-\frac{252843613837367}{53467244989058}a+\frac{161792191955}{22526751628}$, $\frac{13682038459}{106934489978116}a^{15}+\frac{1098548298}{26733622494529}a^{14}-\frac{46221204085}{106934489978116}a^{13}+\frac{7863687699}{106934489978116}a^{12}-\frac{95777703129}{26733622494529}a^{11}-\frac{56991416395}{26733622494529}a^{10}+\frac{27928623585}{26733622494529}a^{9}+\frac{29165786604}{26733622494529}a^{8}+\frac{970606965976}{26733622494529}a^{7}+\frac{517671236110}{26733622494529}a^{6}+\frac{6210578972638}{26733622494529}a^{5}+\frac{3058981002436}{26733622494529}a^{4}+\frac{80535850016667}{106934489978116}a^{3}+\frac{5768540494955}{26733622494529}a^{2}-\frac{80992403584525}{106934489978116}a-\frac{314716467285}{1058757326516}$
| 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: | \( 154432.70553532377 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 154432.70553532377 \cdot 32}{2\cdot\sqrt{723177258444187191407017984}}\cr\approx \mathstrut & 0.223190563793084 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 - 27*x^12 + 44*x^10 + 304*x^8 + 1320*x^6 + 5093*x^4 - 14296*x^2 + 10201)
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^16 - 4*x^14 - 27*x^12 + 44*x^10 + 304*x^8 + 1320*x^6 + 5093*x^4 - 14296*x^2 + 10201, 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^16 - 4*x^14 - 27*x^12 + 44*x^10 + 304*x^8 + 1320*x^6 + 5093*x^4 - 14296*x^2 + 10201);
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^16 - 4*x^14 - 27*x^12 + 44*x^10 + 304*x^8 + 1320*x^6 + 5093*x^4 - 14296*x^2 + 10201);
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
$C_2\times C_8$ (as 16T5):
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)
| An abelian group of order 16 |
| The 16 conjugacy class representatives for $C_8\times C_2$ |
| Character table for $C_8\times C_2$ |
Intermediate fields
| \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-34}) \), \(\Q(\sqrt{-2}, \sqrt{17})\), 4.4.4913.1, 4.0.314432.2, 8.0.98867482624.1, 8.0.105046700288.1, 8.8.1680747204608.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
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 | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.16.1 | $x^{8} + 8 x^{7} + 24 x^{6} + 40 x^{5} + 60 x^{4} + 32 x^{3} + 88 x^{2} + 108$ | $4$ | $2$ | $16$ | $C_4\times C_2$ | $[2, 3]^{2}$ |
| 2.8.16.1 | $x^{8} + 8 x^{7} + 24 x^{6} + 40 x^{5} + 60 x^{4} + 32 x^{3} + 88 x^{2} + 108$ | $4$ | $2$ | $16$ | $C_4\times C_2$ | $[2, 3]^{2}$ | |
|
\(17\)
| 17.8.7.3 | $x^{8} + 17$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |
| 17.8.7.3 | $x^{8} + 17$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |