Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 9*x^14 - 329*x^10 - 1141*x^8 + 1771*x^6 + 16940*x^4 + 25289*x^2 + 14641)
gp: K = bnfinit(y^16 + 9*y^14 - 329*y^10 - 1141*y^8 + 1771*y^6 + 16940*y^4 + 25289*y^2 + 14641, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 9*x^14 - 329*x^10 - 1141*x^8 + 1771*x^6 + 16940*x^4 + 25289*x^2 + 14641);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 9*x^14 - 329*x^10 - 1141*x^8 + 1771*x^6 + 16940*x^4 + 25289*x^2 + 14641)
\( x^{16} + 9x^{14} - 329x^{10} - 1141x^{8} + 1771x^{6} + 16940x^{4} + 25289x^{2} + 14641 \)
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: |
\(3429742096000000000000\)
\(\medspace = 2^{16}\cdot 5^{12}\cdot 11^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(22.18\) | 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: | $2\cdot 5^{3/4}11^{1/2}\approx 22.17960673794632$ | ||
| Ramified primes: |
\(2\), \(5\), \(11\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11}a^{10}-\frac{2}{11}a^{8}+\frac{1}{11}a^{4}+\frac{3}{11}a^{2}$, $\frac{1}{11}a^{11}-\frac{2}{11}a^{9}+\frac{1}{11}a^{5}+\frac{3}{11}a^{3}$, $\frac{1}{121}a^{12}-\frac{2}{121}a^{10}+\frac{2}{11}a^{8}+\frac{34}{121}a^{6}+\frac{58}{121}a^{4}+\frac{4}{11}a^{2}$, $\frac{1}{121}a^{13}-\frac{2}{121}a^{11}+\frac{2}{11}a^{9}+\frac{34}{121}a^{7}+\frac{58}{121}a^{5}+\frac{4}{11}a^{3}$, $\frac{1}{1416999119041}a^{14}-\frac{2280697245}{1416999119041}a^{12}+\frac{5794378428}{128818101731}a^{10}+\frac{125821468884}{1416999119041}a^{8}+\frac{251404430478}{1416999119041}a^{6}+\frac{12214264675}{128818101731}a^{4}+\frac{5841648644}{11710736521}a^{2}+\frac{2132248}{1064612411}$, $\frac{1}{1416999119041}a^{15}-\frac{2280697245}{1416999119041}a^{13}+\frac{5794378428}{128818101731}a^{11}+\frac{125821468884}{1416999119041}a^{9}+\frac{251404430478}{1416999119041}a^{7}+\frac{12214264675}{128818101731}a^{5}+\frac{5841648644}{11710736521}a^{3}+\frac{2132248}{1064612411}a$
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
$C_{4}$, which has order $4$ (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: |
\( \frac{390619951}{1416999119041} a^{14} + \frac{3017044885}{1416999119041} a^{12} - \frac{482992141}{128818101731} a^{10} - \frac{120232777966}{1416999119041} a^{8} - \frac{233436025021}{1416999119041} a^{6} + \frac{103506700392}{128818101731} a^{4} + \frac{36413729521}{11710736521} a^{2} + \frac{2283371231}{1064612411} \)
(order $10$)
| 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: |
$\frac{6449642}{1416999119041}a^{14}-\frac{257656005}{1416999119041}a^{12}-\frac{10136986}{128818101731}a^{10}+\frac{14824619811}{1416999119041}a^{8}+\frac{30231538439}{1416999119041}a^{6}-\frac{25958759454}{128818101731}a^{4}-\frac{4548099261}{11710736521}a^{2}+\frac{637742329}{1064612411}$, $\frac{15605842}{128818101731}a^{14}+\frac{50565124}{128818101731}a^{12}-\frac{327661032}{128818101731}a^{10}-\frac{2742048992}{128818101731}a^{8}-\frac{170999912}{11710736521}a^{6}+\frac{15487288001}{128818101731}a^{4}+\frac{8189490295}{11710736521}a^{2}+\frac{1063233011}{1064612411}$, $\frac{411588495}{1416999119041}a^{14}+\frac{3000660885}{1416999119041}a^{12}-\frac{61943859}{11710736521}a^{10}-\frac{125456493823}{1416999119041}a^{8}-\frac{207349484235}{1416999119041}a^{6}+\frac{118597453451}{128818101731}a^{4}+\frac{33467961743}{11710736521}a^{2}+\frac{1891953376}{1064612411}$, $\frac{143980108}{1416999119041}a^{15}-\frac{531936}{1416999119041}a^{14}+\frac{584799771}{1416999119041}a^{13}-\frac{490913196}{1416999119041}a^{12}-\frac{274669785}{128818101731}a^{11}-\frac{311072094}{128818101731}a^{10}-\frac{28913801667}{1416999119041}a^{9}+\frac{21699282398}{1416999119041}a^{8}+\frac{13201805410}{1416999119041}a^{7}+\frac{180421747974}{1416999119041}a^{6}+\frac{31155922713}{128818101731}a^{5}+\frac{9601256634}{128818101731}a^{4}+\frac{4524849588}{11710736521}a^{3}-\frac{22317015492}{11710736521}a^{2}-\frac{983637286}{1064612411}a-\frac{2536479235}{1064612411}$, $\frac{280424283}{1416999119041}a^{15}-\frac{159668689}{1416999119041}a^{14}+\frac{2321314124}{1416999119041}a^{13}+\frac{488751821}{1416999119041}a^{12}-\frac{260321473}{128818101731}a^{11}+\frac{908469381}{128818101731}a^{10}-\frac{90790965038}{1416999119041}a^{9}+\frac{6925882938}{1416999119041}a^{8}-\frac{204067951913}{1416999119041}a^{7}-\frac{253276571197}{1416999119041}a^{6}+\frac{76087067536}{128818101731}a^{5}-\frac{43825916864}{128818101731}a^{4}+\frac{26892239387}{11710736521}a^{3}+\frac{14212585633}{11710736521}a^{2}+\frac{943614500}{1064612411}a+\frac{2353968051}{1064612411}$, $\frac{512121492}{1416999119041}a^{15}+\frac{12815896}{1416999119041}a^{14}+\frac{3807383090}{1416999119041}a^{13}-\frac{94546990}{1416999119041}a^{12}-\frac{402962642}{128818101731}a^{11}+\frac{146391191}{128818101731}a^{10}-\frac{152860843132}{1416999119041}a^{9}+\frac{5751727118}{1416999119041}a^{8}-\frac{370090746962}{1416999119041}a^{7}+\frac{16483337732}{1416999119041}a^{6}+\frac{112822157361}{128818101731}a^{5}-\frac{11354463202}{128818101731}a^{4}+\frac{54641725322}{11710736521}a^{3}-\frac{2050872768}{11710736521}a^{2}+\frac{3464785782}{1064612411}a-\frac{1931976162}{1064612411}$, $\frac{27758530}{1416999119041}a^{15}-\frac{282971196}{1416999119041}a^{14}+\frac{851629158}{1416999119041}a^{13}-\frac{966632817}{1416999119041}a^{12}+\frac{80710004}{128818101731}a^{11}+\frac{57613152}{128818101731}a^{10}-\frac{23497371087}{1416999119041}a^{9}+\frac{39053858061}{1416999119041}a^{8}-\frac{64131697650}{1416999119041}a^{7}+\frac{110297152938}{1416999119041}a^{6}+\frac{24261224652}{128818101731}a^{5}-\frac{1209072658}{128818101731}a^{4}+\frac{2088521230}{11710736521}a^{3}-\frac{15752744049}{11710736521}a^{2}-\frac{121526516}{1064612411}a-\frac{1005856413}{1064612411}$
| 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: | \( 13401.1589099 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 13401.1589099 \cdot 4}{10\cdot\sqrt{3429742096000000000000}}\cr\approx \mathstrut & 0.222336396857 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 + 9*x^14 - 329*x^10 - 1141*x^8 + 1771*x^6 + 16940*x^4 + 25289*x^2 + 14641)
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 + 9*x^14 - 329*x^10 - 1141*x^8 + 1771*x^6 + 16940*x^4 + 25289*x^2 + 14641, 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 + 9*x^14 - 329*x^10 - 1141*x^8 + 1771*x^6 + 16940*x^4 + 25289*x^2 + 14641);
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 + 9*x^14 - 329*x^10 - 1141*x^8 + 1771*x^6 + 16940*x^4 + 25289*x^2 + 14641);
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 16T21):
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 20 conjugacy class representatives for $C_2 \times (C_2^2:C_4)$ |
| Character table for $C_2 \times (C_2^2:C_4)$ |
Intermediate fields
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 16 siblings: | 16.8.3429742096000000000000.1, 16.0.234256000000000000.1, 16.0.3429742096000000000000.2 |
| Minimal sibling: | 16.0.234256000000000000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ |
| 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
|
\(5\)
| 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
|
\(11\)
| 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |