Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 70*x^12 - 56*x^11 + 144*x^10 - 588*x^9 + 1190*x^8 - 1276*x^7 + 754*x^6 - 232*x^5 + 61*x^4 - 64*x^3 + 48*x^2 - 16*x + 2)
gp: K = bnfinit(y^16 - 8*y^15 + 28*y^14 - 56*y^13 + 70*y^12 - 56*y^11 + 144*y^10 - 588*y^9 + 1190*y^8 - 1276*y^7 + 754*y^6 - 232*y^5 + 61*y^4 - 64*y^3 + 48*y^2 - 16*y + 2, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 70*x^12 - 56*x^11 + 144*x^10 - 588*x^9 + 1190*x^8 - 1276*x^7 + 754*x^6 - 232*x^5 + 61*x^4 - 64*x^3 + 48*x^2 - 16*x + 2);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 70*x^12 - 56*x^11 + 144*x^10 - 588*x^9 + 1190*x^8 - 1276*x^7 + 754*x^6 - 232*x^5 + 61*x^4 - 64*x^3 + 48*x^2 - 16*x + 2)
\( x^{16} - 8 x^{15} + 28 x^{14} - 56 x^{13} + 70 x^{12} - 56 x^{11} + 144 x^{10} - 588 x^{9} + 1190 x^{8} + \cdots + 2 \)
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: |
\(119842600295694598144\)
\(\medspace = 2^{34}\cdot 17^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.99\) | 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: | $2^{11/4}17^{1/2}\approx 27.736837922354518$ | ||
| 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: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{7}a^{8}+\frac{3}{7}a^{7}-\frac{1}{7}a^{6}+\frac{3}{7}a^{5}+\frac{1}{7}a^{4}+\frac{2}{7}a^{2}-\frac{2}{7}a-\frac{3}{7}$, $\frac{1}{7}a^{9}-\frac{3}{7}a^{7}-\frac{1}{7}a^{6}-\frac{1}{7}a^{5}-\frac{3}{7}a^{4}+\frac{2}{7}a^{3}-\frac{1}{7}a^{2}+\frac{3}{7}a+\frac{2}{7}$, $\frac{1}{7}a^{10}+\frac{1}{7}a^{7}+\frac{3}{7}a^{6}-\frac{1}{7}a^{5}-\frac{2}{7}a^{4}-\frac{1}{7}a^{3}+\frac{2}{7}a^{2}+\frac{3}{7}a-\frac{2}{7}$, $\frac{1}{7}a^{11}+\frac{2}{7}a^{5}-\frac{2}{7}a^{4}+\frac{2}{7}a^{3}+\frac{1}{7}a^{2}+\frac{3}{7}$, $\frac{1}{14}a^{12}-\frac{1}{14}a^{10}+\frac{3}{7}a^{7}-\frac{1}{14}a^{6}+\frac{3}{7}a^{5}-\frac{3}{14}a^{4}+\frac{1}{7}a^{3}-\frac{1}{7}a^{2}+\frac{1}{7}$, $\frac{1}{14}a^{13}-\frac{1}{14}a^{11}-\frac{5}{14}a^{7}-\frac{1}{7}a^{6}-\frac{1}{2}a^{5}-\frac{2}{7}a^{4}-\frac{1}{7}a^{3}+\frac{1}{7}a^{2}+\frac{2}{7}$, $\frac{1}{182}a^{14}+\frac{3}{91}a^{13}-\frac{1}{182}a^{12}+\frac{3}{91}a^{11}-\frac{3}{91}a^{10}-\frac{6}{91}a^{9}-\frac{9}{182}a^{8}+\frac{5}{13}a^{7}-\frac{9}{26}a^{6}-\frac{29}{91}a^{5}-\frac{38}{91}a^{4}-\frac{23}{91}a^{3}-\frac{34}{91}a^{2}-\frac{4}{13}a+\frac{16}{91}$, $\frac{1}{182}a^{15}+\frac{1}{91}a^{13}-\frac{1}{182}a^{12}-\frac{3}{182}a^{11}+\frac{11}{182}a^{10}+\frac{11}{182}a^{9}-\frac{3}{91}a^{8}+\frac{38}{91}a^{7}-\frac{5}{182}a^{6}-\frac{53}{182}a^{5}+\frac{33}{182}a^{4}-\frac{45}{91}a^{2}+\frac{15}{91}a-\frac{18}{91}$
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
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: | $7$ | 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: |
$\frac{549}{26}a^{15}-\frac{1095}{7}a^{14}+\frac{6490}{13}a^{13}-\frac{23169}{26}a^{12}+\frac{175307}{182}a^{11}-\frac{115261}{182}a^{10}+\frac{488849}{182}a^{9}-\frac{986178}{91}a^{8}+\frac{1709180}{91}a^{7}-\frac{2930981}{182}a^{6}+\frac{1243013}{182}a^{5}-\frac{212377}{182}a^{4}+\frac{4703}{7}a^{3}-\frac{85796}{91}a^{2}+\frac{42773}{91}a-\frac{1068}{13}$, $\frac{3403}{26}a^{15}-\frac{25591}{26}a^{14}+\frac{581011}{182}a^{13}-\frac{150725}{26}a^{12}+\frac{580471}{91}a^{11}-\frac{388298}{91}a^{10}+\frac{436773}{26}a^{9}-\frac{12539571}{182}a^{8}+\frac{3189703}{26}a^{7}-\frac{19675493}{182}a^{6}+\frac{4256988}{91}a^{5}-\frac{720303}{91}a^{4}+\frac{54462}{13}a^{3}-\frac{578580}{91}a^{2}+\frac{293362}{91}a-\frac{7091}{13}$, $\frac{96608}{91}a^{15}-\frac{724560}{91}a^{14}+\frac{2342853}{91}a^{13}-\frac{652213}{14}a^{12}+\frac{4645128}{91}a^{11}-\frac{883179}{26}a^{10}+\frac{12369381}{91}a^{9}-\frac{50622570}{91}a^{8}+\frac{89665294}{91}a^{7}-\frac{156979503}{182}a^{6}+\frac{33673249}{91}a^{5}-\frac{11252971}{182}a^{4}+\frac{3092111}{91}a^{3}-\frac{4636976}{91}a^{2}+\frac{2322386}{91}a-\frac{388078}{91}$, $\frac{1137}{91}a^{15}-\frac{8683}{91}a^{14}+\frac{8197}{26}a^{13}-\frac{53294}{91}a^{12}+\frac{120655}{182}a^{11}-\frac{41852}{91}a^{10}+\frac{21220}{13}a^{9}-\frac{614555}{91}a^{8}+\frac{2261271}{182}a^{7}-\frac{1042511}{91}a^{6}+\frac{137193}{26}a^{5}-\frac{88512}{91}a^{4}+\frac{36133}{91}a^{3}-\frac{59485}{91}a^{2}+\frac{33105}{91}a-\frac{6099}{91}$, $\frac{1137}{91}a^{15}-92a^{14}+\frac{7575}{26}a^{13}-\frac{93703}{182}a^{12}+\frac{99947}{182}a^{11}-\frac{64543}{182}a^{10}+\frac{143664}{91}a^{9}-\frac{577405}{91}a^{8}+\frac{1972687}{182}a^{7}-\frac{1654609}{182}a^{6}+\frac{679199}{182}a^{5}-\frac{112145}{182}a^{4}+\frac{2860}{7}a^{3}-\frac{49459}{91}a^{2}+\frac{3311}{13}a-\frac{536}{13}$, $\frac{1133}{182}a^{14}-\frac{1133}{26}a^{13}+\frac{11751}{91}a^{12}-\frac{37909}{182}a^{11}+\frac{35361}{182}a^{10}-\frac{9164}{91}a^{9}+\frac{135689}{182}a^{8}-\frac{75107}{26}a^{7}+\frac{393824}{91}a^{6}-\frac{523041}{182}a^{5}+\frac{128535}{182}a^{4}+\frac{643}{91}a^{3}+\frac{18171}{91}a^{2}-\frac{18269}{91}a+\frac{4530}{91}$, $\frac{2601}{182}a^{14}-\frac{2601}{26}a^{13}+\frac{53975}{182}a^{12}-\frac{87159}{182}a^{11}+\frac{40752}{91}a^{10}-\frac{21248}{91}a^{9}+\frac{311783}{182}a^{8}-\frac{1207019}{182}a^{7}+\frac{1810707}{182}a^{6}-\frac{1208369}{182}a^{5}+\frac{21634}{13}a^{4}+\frac{55}{91}a^{3}+\frac{5925}{13}a^{2}-\frac{41628}{91}a+\frac{10429}{91}$
| 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: | \( 11218.0567044 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 11218.0567044 \cdot 1}{2\cdot\sqrt{119842600295694598144}}\cr\approx \mathstrut & 1.24457470581 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 70*x^12 - 56*x^11 + 144*x^10 - 588*x^9 + 1190*x^8 - 1276*x^7 + 754*x^6 - 232*x^5 + 61*x^4 - 64*x^3 + 48*x^2 - 16*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^16 - 8*x^15 + 28*x^14 - 56*x^13 + 70*x^12 - 56*x^11 + 144*x^10 - 588*x^9 + 1190*x^8 - 1276*x^7 + 754*x^6 - 232*x^5 + 61*x^4 - 64*x^3 + 48*x^2 - 16*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^16 - 8*x^15 + 28*x^14 - 56*x^13 + 70*x^12 - 56*x^11 + 144*x^10 - 588*x^9 + 1190*x^8 - 1276*x^7 + 754*x^6 - 232*x^5 + 61*x^4 - 64*x^3 + 48*x^2 - 16*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^16 - 8*x^15 + 28*x^14 - 56*x^13 + 70*x^12 - 56*x^11 + 144*x^10 - 588*x^9 + 1190*x^8 - 1276*x^7 + 754*x^6 - 232*x^5 + 61*x^4 - 64*x^3 + 48*x^2 - 16*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
$C_2^3:C_4$ (as 16T33):
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 32 |
| The 11 conjugacy class representatives for $C_2^3:C_4$ |
| Character table for $C_2^3:C_4$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{17}) \), 4.0.2312.1 x2, \(\Q(\sqrt{2}, \sqrt{17})\), 4.0.1088.2 x2, 8.0.1368408064.2 x2, 8.4.10947264512.2 x2, 8.0.342102016.2 |
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]
Sibling fields
| Galois closure: | deg 32 |
| Degree 8 siblings: | 8.4.350312464384.4, 8.0.1368408064.2, 8.4.10947264512.2, 8.0.1212153856.11 |
| Degree 16 siblings: | deg 16, deg 16, deg 16 |
| Minimal sibling: | 8.0.1368408064.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.2.0.1}{2} }^{8}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.11.3 | $x^{4} + 4 x^{2} + 18$ | $4$ | $1$ | $11$ | $C_4$ | $[3, 4]$ |
| 2.4.11.3 | $x^{4} + 4 x^{2} + 18$ | $4$ | $1$ | $11$ | $C_4$ | $[3, 4]$ | |
| 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
|
\(17\)
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |