Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 476*x^14 + 68306*x^12 + 4338264*x^10 + 133226688*x^8 + 1942889200*x^6 + 13328219912*x^4 + 38080628320*x^2 + 26656439824)
gp: K = bnfinit(y^16 + 476*y^14 + 68306*y^12 + 4338264*y^10 + 133226688*y^8 + 1942889200*y^6 + 13328219912*y^4 + 38080628320*y^2 + 26656439824, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 476*x^14 + 68306*x^12 + 4338264*x^10 + 133226688*x^8 + 1942889200*x^6 + 13328219912*x^4 + 38080628320*x^2 + 26656439824);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 476*x^14 + 68306*x^12 + 4338264*x^10 + 133226688*x^8 + 1942889200*x^6 + 13328219912*x^4 + 38080628320*x^2 + 26656439824)
\( x^{16} + 476 x^{14} + 68306 x^{12} + 4338264 x^{10} + 133226688 x^{8} + 1942889200 x^{6} + \cdots + 26656439824 \)
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: |
\(17076113336960240702301670118118129664\)
\(\medspace = 2^{44}\cdot 7^{8}\cdot 17^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(212.34\) | 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: | $2^{11/4}7^{1/2}17^{7/8}\approx 212.3363336696819$ | ||
| Ramified primes: |
\(2\), \(7\), \(17\)
| 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: | \(1904=2^{4}\cdot 7\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1904}(897,·)$, $\chi_{1904}(1413,·)$, $\chi_{1904}(1,·)$, $\chi_{1904}(909,·)$, $\chi_{1904}(461,·)$, $\chi_{1904}(1749,·)$, $\chi_{1904}(953,·)$, $\chi_{1904}(1177,·)$, $\chi_{1904}(349,·)$, $\chi_{1904}(1861,·)$, $\chi_{1904}(225,·)$, $\chi_{1904}(1121,·)$, $\chi_{1904}(169,·)$, $\chi_{1904}(797,·)$, $\chi_{1904}(1849,·)$, $\chi_{1904}(1301,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{7}a^{2}$, $\frac{1}{7}a^{3}$, $\frac{1}{98}a^{4}$, $\frac{1}{98}a^{5}$, $\frac{1}{686}a^{6}$, $\frac{1}{686}a^{7}$, $\frac{1}{163268}a^{8}$, $\frac{1}{163268}a^{9}$, $\frac{1}{1142876}a^{10}$, $\frac{1}{1142876}a^{11}$, $\frac{1}{1664027456}a^{12}-\frac{1}{3714347}a^{10}-\frac{1}{1061242}a^{8}+\frac{5}{35672}a^{6}-\frac{3}{637}a^{4}-\frac{1}{91}a^{2}-\frac{23}{104}$, $\frac{1}{1664027456}a^{13}-\frac{1}{3714347}a^{11}-\frac{1}{1061242}a^{9}+\frac{5}{35672}a^{7}-\frac{3}{637}a^{5}-\frac{1}{91}a^{3}-\frac{23}{104}a$, $\frac{1}{1036689105088}a^{14}-\frac{29}{148098443584}a^{12}+\frac{313}{1322307532}a^{10}-\frac{757}{377802152}a^{8}+\frac{47}{3174808}a^{6}+\frac{37}{56693}a^{4}-\frac{823}{64792}a^{2}-\frac{285}{9256}$, $\frac{1}{1036689105088}a^{15}-\frac{29}{148098443584}a^{13}+\frac{313}{1322307532}a^{11}-\frac{757}{377802152}a^{9}+\frac{47}{3174808}a^{7}+\frac{37}{56693}a^{5}-\frac{823}{64792}a^{3}-\frac{285}{9256}a$
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
$C_{2}\times C_{2}\times C_{2}\times C_{6}\times C_{233382}$, which has order $11202336$ (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: |
$\frac{29}{19936328944}a^{14}+\frac{137}{203431928}a^{12}+\frac{2313}{25428991}a^{10}+\frac{38389}{7265426}a^{8}+\frac{8661}{61054}a^{6}+\frac{1020}{623}a^{4}+\frac{8885}{1246}a^{2}+\frac{718}{89}$, $\frac{101}{74049221792}a^{14}+\frac{48561}{74049221792}a^{12}+\frac{9007}{94450538}a^{10}+\frac{4743}{793702}a^{8}+\frac{38211}{226772}a^{6}+\frac{15217}{8099}a^{4}+\frac{33645}{4628}a^{2}+\frac{24209}{4628}$, $\frac{2097}{1036689105088}a^{14}+\frac{140683}{148098443584}a^{12}+\frac{174337}{1322307532}a^{10}+\frac{24845}{3174808}a^{8}+\frac{661751}{3174808}a^{6}+\frac{124492}{56693}a^{4}+\frac{75887}{9256}a^{2}+\frac{53123}{9256}$, $\frac{683}{1036689105088}a^{14}+\frac{127}{431773888}a^{12}+\frac{48239}{1322307532}a^{10}+\frac{839}{453544}a^{8}+\frac{126797}{3174808}a^{6}+\frac{17973}{56693}a^{4}+\frac{8597}{9256}a^{2}+\frac{4705}{9256}$, $\frac{141}{18512305448}a^{14}+\frac{527249}{148098443584}a^{12}+\frac{323741}{661153766}a^{10}+\frac{2722235}{94450538}a^{8}+\frac{2430661}{3174808}a^{6}+\frac{463566}{56693}a^{4}+\frac{252584}{8099}a^{2}+\frac{219657}{9256}$, $\frac{951}{1036689105088}a^{14}+\frac{1851}{4355836576}a^{12}+\frac{1454}{25428991}a^{10}+\frac{1218157}{377802152}a^{8}+\frac{123675}{1587404}a^{6}+\frac{35543}{56693}a^{4}+\frac{38975}{64792}a^{2}-\frac{24045}{4628}$, $\frac{4395}{1036689105088}a^{14}+\frac{293515}{148098443584}a^{12}+\frac{360501}{1322307532}a^{10}+\frac{6059281}{377802152}a^{8}+\frac{1348791}{3174808}a^{6}+\frac{254552}{56693}a^{4}+\frac{1077131}{64792}a^{2}+\frac{9695}{712}$
| 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: | \( 103646.40189541418 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 103646.40189541418 \cdot 11202336}{2\cdot\sqrt{17076113336960240702301670118118129664}}\cr\approx \mathstrut & 0.341253615728791 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 + 476*x^14 + 68306*x^12 + 4338264*x^10 + 133226688*x^8 + 1942889200*x^6 + 13328219912*x^4 + 38080628320*x^2 + 26656439824)
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 + 476*x^14 + 68306*x^12 + 4338264*x^10 + 133226688*x^8 + 1942889200*x^6 + 13328219912*x^4 + 38080628320*x^2 + 26656439824, 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 + 476*x^14 + 68306*x^12 + 4338264*x^10 + 133226688*x^8 + 1942889200*x^6 + 13328219912*x^4 + 38080628320*x^2 + 26656439824);
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 + 476*x^14 + 68306*x^12 + 4338264*x^10 + 133226688*x^8 + 1942889200*x^6 + 13328219912*x^4 + 38080628320*x^2 + 26656439824);
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
| \(\Q(\sqrt{2}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{2}, \sqrt{17})\), 4.4.314432.1, 4.4.4913.1, 8.8.98867482624.1, 8.0.4132325415182139392.2, 8.0.4132325415182139392.1 |
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 | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.1.0.1}{1} }^{16}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.22.5 | $x^{8} + 20 x^{7} + 154 x^{6} + 592 x^{5} + 1315 x^{4} + 1968 x^{3} + 2050 x^{2} + 828 x + 111$ | $4$ | $2$ | $22$ | $C_4\times C_2$ | $[3, 4]^{2}$ |
| 2.8.22.5 | $x^{8} + 20 x^{7} + 154 x^{6} + 592 x^{5} + 1315 x^{4} + 1968 x^{3} + 2050 x^{2} + 828 x + 111$ | $4$ | $2$ | $22$ | $C_4\times C_2$ | $[3, 4]^{2}$ | |
|
\(7\)
| 7.8.4.2 | $x^{8} + 245 x^{4} - 1372 x^{2} + 7203$ | $2$ | $4$ | $4$ | $C_8$ | $[\ ]_{2}^{4}$ |
| 7.8.4.2 | $x^{8} + 245 x^{4} - 1372 x^{2} + 7203$ | $2$ | $4$ | $4$ | $C_8$ | $[\ ]_{2}^{4}$ | |
|
\(17\)
| 17.8.7.1 | $x^{8} + 68$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |
| 17.8.7.1 | $x^{8} + 68$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |