Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 40*x^12 + 144*x^10 - 236*x^8 + 176*x^6 - 48*x^4 + 2)
gp: K = bnfinit(y^16 - 40*y^12 + 144*y^10 - 236*y^8 + 176*y^6 - 48*y^4 + 2, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 40*x^12 + 144*x^10 - 236*x^8 + 176*x^6 - 48*x^4 + 2);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 40*x^12 + 144*x^10 - 236*x^8 + 176*x^6 - 48*x^4 + 2)
\( x^{16} - 40x^{12} + 144x^{10} - 236x^{8} + 176x^{6} - 48x^{4} + 2 \)
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: | $[4, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(604462909807314587353088\)
\(\medspace = 2^{79}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(30.64\) | 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: | $2^{2849/512}\approx 47.32245861429085$ | ||
| Ramified primes: |
\(2\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{399889}a^{14}+\frac{75071}{399889}a^{12}+\frac{19324}{399889}a^{10}-\frac{125144}{399889}a^{8}-\frac{93183}{399889}a^{6}-\frac{82540}{399889}a^{4}-\frac{80333}{399889}a^{2}+\frac{47366}{399889}$, $\frac{1}{399889}a^{15}+\frac{75071}{399889}a^{13}+\frac{19324}{399889}a^{11}-\frac{125144}{399889}a^{9}-\frac{93183}{399889}a^{7}-\frac{82540}{399889}a^{5}-\frac{80333}{399889}a^{3}+\frac{47366}{399889}a$
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
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: | $9$ | 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: |
$\frac{31434}{399889}a^{14}+\frac{36825}{399889}a^{12}-\frac{1200442}{399889}a^{10}+\frac{3130709}{399889}a^{8}-\frac{4326276}{399889}a^{6}+\frac{1916917}{399889}a^{4}-\frac{288376}{399889}a^{2}+\frac{116097}{399889}$, $\frac{644}{8161}a^{14}-\frac{40}{8161}a^{12}-\frac{25352}{8161}a^{10}+\frac{95071}{8161}a^{8}-\frac{173400}{8161}a^{6}+\frac{151892}{8161}a^{4}-\frac{59000}{8161}a^{2}+\frac{6047}{8161}$, $\frac{122}{399889}a^{14}-\frac{38785}{399889}a^{12}-\frac{41806}{399889}a^{10}+\frac{1527770}{399889}a^{8}-\frac{4170324}{399889}a^{6}+\frac{5525791}{399889}a^{4}-\frac{2602624}{399889}a^{2}+\frac{580095}{399889}$, $\frac{110340}{399889}a^{14}+\frac{33394}{399889}a^{12}-\frac{4396767}{399889}a^{10}+\frac{14574103}{399889}a^{8}-\frac{21860147}{399889}a^{6}+\frac{13204712}{399889}a^{4}-\frac{2003091}{399889}a^{2}+\frac{215099}{399889}$, $\frac{4666}{399889}a^{14}-\frac{21478}{399889}a^{12}-\frac{209130}{399889}a^{10}+\frac{1515592}{399889}a^{8}-\frac{3311647}{399889}a^{6}+\frac{3160579}{399889}a^{4}-\frac{537674}{399889}a^{2}-\frac{128861}{399889}$, $\frac{122357}{399889}a^{15}-\frac{122357}{399889}a^{14}+\frac{12017}{399889}a^{13}-\frac{12017}{399889}a^{12}-\frac{4915657}{399889}a^{11}+\frac{4915657}{399889}a^{10}+\frac{17100518}{399889}a^{9}-\frac{17100518}{399889}a^{8}-\frac{26349837}{399889}a^{7}+\frac{26349837}{399889}a^{6}+\frac{17045142}{399889}a^{5}-\frac{17045142}{399889}a^{4}-\frac{2032706}{399889}a^{3}+\frac{2032706}{399889}a^{2}-\frac{29615}{399889}a+\frac{29615}{399889}$, $\frac{25196}{399889}a^{15}-\frac{15528}{399889}a^{14}+\frac{13946}{399889}a^{13}-\frac{26053}{399889}a^{12}-\frac{977076}{399889}a^{11}+\frac{653456}{399889}a^{10}+\frac{3195653}{399889}a^{9}-\frac{1024286}{399889}a^{8}-\frac{4889217}{399889}a^{7}-\frac{1052445}{399889}a^{6}+\frac{944627}{399889}a^{5}+\frac{1636431}{399889}a^{4}+\frac{967628}{399889}a^{3}-\frac{242856}{399889}a^{2}-\frac{234929}{399889}a-\frac{103377}{399889}$, $\frac{338249}{399889}a^{15}-\frac{226071}{399889}a^{14}+\frac{139068}{399889}a^{13}-\frac{86881}{399889}a^{12}-\frac{13458255}{399889}a^{11}+\frac{8988879}{399889}a^{10}+\frac{43205362}{399889}a^{9}-\frac{29109645}{399889}a^{8}-\frac{62588160}{399889}a^{7}+\frac{43009485}{399889}a^{6}+\frac{34768196}{399889}a^{5}-\frac{25712963}{399889}a^{4}-\frac{2898590}{399889}a^{3}+\frac{4001598}{399889}a^{2}-\frac{1250318}{399889}a+\frac{548545}{399889}$, $\frac{266219}{399889}a^{15}+\frac{644}{8161}a^{14}+\frac{73996}{399889}a^{13}-\frac{40}{8161}a^{12}-\frac{10553143}{399889}a^{11}-\frac{25352}{8161}a^{10}+\frac{35431953}{399889}a^{9}+\frac{95071}{8161}a^{8}-\frac{55955422}{399889}a^{7}-\frac{173400}{8161}a^{6}+\frac{40972968}{399889}a^{5}+\frac{151892}{8161}a^{4}-\frac{15702878}{399889}a^{3}-\frac{59000}{8161}a^{2}+\frac{4028207}{399889}a-\frac{10275}{8161}$
| 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: | \( 590539.358696 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 590539.358696 \cdot 1}{2\cdot\sqrt{604462909807314587353088}}\cr\approx \mathstrut & 0.373880845812 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 40*x^12 + 144*x^10 - 236*x^8 + 176*x^6 - 48*x^4 + 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 - 40*x^12 + 144*x^10 - 236*x^8 + 176*x^6 - 48*x^4 + 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 - 40*x^12 + 144*x^10 - 236*x^8 + 176*x^6 - 48*x^4 + 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 - 40*x^12 + 144*x^10 - 236*x^8 + 176*x^6 - 48*x^4 + 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
$C_2^7.C_8$ (as 16T1155):
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 1024 |
| The 40 conjugacy class representatives for $C_2^7.C_8$ |
| Character table for $C_2^7.C_8$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.4.2147483648.2 |
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
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $16$ | $16$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ | $16$ | $16$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | $16$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | $16$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | $16$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | $16$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | $16$ | $16$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.16.79.2518 | $x^{16} + 56 x^{12} + 32 x^{11} + 4 x^{8} + 32 x^{7} + 48 x^{6} + 32 x^{5} + 32 x^{2} + 34$ | $16$ | $1$ | $79$ | 16T1155 | $[2, 3, 7/2, 4, 17/4, 19/4, 5, 41/8, 43/8, 6]$ |