Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 24*x^12 + 110*x^10 + 176*x^8 + 440*x^6 + 989*x^4 + 880*x^2 + 256)
gp: K = bnfinit(y^16 + 24*y^12 + 110*y^10 + 176*y^8 + 440*y^6 + 989*y^4 + 880*y^2 + 256, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 24*x^12 + 110*x^10 + 176*x^8 + 440*x^6 + 989*x^4 + 880*x^2 + 256);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 24*x^12 + 110*x^10 + 176*x^8 + 440*x^6 + 989*x^4 + 880*x^2 + 256)
\( x^{16} + 24x^{12} + 110x^{10} + 176x^{8} + 440x^{6} + 989x^{4} + 880x^{2} + 256 \)
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: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{4}a^{7}+\frac{1}{4}a$, $\frac{1}{4}a^{8}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{9}+\frac{1}{4}a^{3}$, $\frac{1}{4}a^{10}+\frac{1}{4}a^{4}$, $\frac{1}{4}a^{11}+\frac{1}{4}a^{5}$, $\frac{1}{28}a^{12}+\frac{3}{28}a^{10}-\frac{1}{28}a^{8}+\frac{5}{28}a^{6}-\frac{13}{28}a^{4}+\frac{1}{4}a^{2}-\frac{1}{7}$, $\frac{1}{28}a^{13}+\frac{3}{28}a^{11}-\frac{1}{28}a^{9}-\frac{1}{14}a^{7}-\frac{13}{28}a^{5}+\frac{1}{4}a^{3}-\frac{11}{28}a$, $\frac{1}{707728}a^{14}+\frac{81}{25276}a^{12}-\frac{1557}{88466}a^{10}-\frac{8403}{353864}a^{8}-\frac{8111}{25276}a^{6}-\frac{19729}{88466}a^{4}-\frac{230759}{707728}a^{2}-\frac{10788}{44233}$, $\frac{1}{2830912}a^{15}-\frac{719}{88466}a^{13}+\frac{1583}{50552}a^{11}-\frac{24671}{202208}a^{9}-\frac{22093}{176932}a^{7}+\frac{43461}{353864}a^{5}-\frac{53827}{2830912}a^{3}+\frac{83997}{176932}a$
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
$C_{4}$, which has order $4$
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: |
$\frac{891}{101104}a^{14}-\frac{323}{25276}a^{12}+\frac{2893}{12638}a^{10}+\frac{32537}{50552}a^{8}+\frac{14245}{25276}a^{6}+\frac{38833}{12638}a^{4}+\frac{418407}{101104}a^{2}-\frac{909}{6319}$, $\frac{195749}{2830912}a^{15}-\frac{7207}{176932}a^{13}+\frac{590801}{353864}a^{11}+\frac{9422715}{1415456}a^{9}+\frac{693445}{88466}a^{7}+\frac{8923085}{353864}a^{5}+\frac{149403649}{2830912}a^{3}+\frac{4537121}{176932}a$, $\frac{115317}{2830912}a^{15}-\frac{662}{44233}a^{13}+\frac{340723}{353864}a^{11}+\frac{5924483}{1415456}a^{9}+\frac{443459}{88466}a^{7}+\frac{5488559}{353864}a^{5}+\frac{95857409}{2830912}a^{3}+\frac{1520065}{88466}a$, $\frac{4587}{25276}a^{15}+\frac{891}{101104}a^{14}-\frac{4077}{25276}a^{13}-\frac{323}{25276}a^{12}+\frac{28617}{6319}a^{11}+\frac{2893}{12638}a^{10}+\frac{400775}{25276}a^{9}+\frac{32537}{50552}a^{8}+\frac{236775}{12638}a^{7}+\frac{14245}{25276}a^{6}+\frac{405687}{6319}a^{5}+\frac{38833}{12638}a^{4}+\frac{779856}{6319}a^{3}+\frac{418407}{101104}a^{2}+\frac{1490131}{25276}a+\frac{5410}{6319}$, $\frac{28039}{1415456}a^{15}+\frac{41667}{353864}a^{14}+\frac{1457}{88466}a^{13}-\frac{7871}{88466}a^{12}+\frac{70757}{176932}a^{11}+\frac{128034}{44233}a^{10}+\frac{1944769}{707728}a^{9}+\frac{1903689}{176932}a^{8}+\frac{276617}{88466}a^{7}+\frac{1117381}{88466}a^{6}+\frac{1449837}{176932}a^{5}+\frac{267647}{6319}a^{4}+\frac{32176331}{1415456}a^{3}+\frac{29704999}{353864}a^{2}+\frac{483143}{44233}a+\frac{260447}{6319}$, $\frac{90369}{2830912}a^{15}-\frac{41667}{353864}a^{14}-\frac{387}{176932}a^{13}+\frac{7871}{88466}a^{12}+\frac{259719}{353864}a^{11}-\frac{128034}{44233}a^{10}+\frac{5013447}{1415456}a^{9}-\frac{1903689}{176932}a^{8}+\frac{787203}{176932}a^{7}-\frac{1117381}{88466}a^{6}+\frac{4401235}{353864}a^{5}-\frac{267647}{6319}a^{4}+\frac{84142013}{2830912}a^{3}-\frac{29704999}{353864}a^{2}+\frac{1532791}{88466}a-\frac{254128}{6319}$, $\frac{370305}{2830912}a^{15}-\frac{14811}{707728}a^{14}-\frac{15417}{176932}a^{13}+\frac{919}{25276}a^{12}+\frac{1130733}{353864}a^{11}-\frac{102031}{176932}a^{10}+\frac{17382143}{1415456}a^{9}-\frac{457159}{353864}a^{8}+\frac{651873}{44233}a^{7}-\frac{42721}{25276}a^{6}+\frac{2428647}{50552}a^{5}-\frac{1276715}{176932}a^{4}+\frac{276765805}{2830912}a^{3}-\frac{6216515}{707728}a^{2}+\frac{1215411}{25276}a-\frac{121227}{44233}$
| 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: | \( 4962.2860285 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 4962.2860285 \cdot 4}{2\cdot\sqrt{3429742096000000000000}}\cr\approx \mathstrut & 0.41164230764 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 + 24*x^12 + 110*x^10 + 176*x^8 + 440*x^6 + 989*x^4 + 880*x^2 + 256)
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 + 24*x^12 + 110*x^10 + 176*x^8 + 440*x^6 + 989*x^4 + 880*x^2 + 256, 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 + 24*x^12 + 110*x^10 + 176*x^8 + 440*x^6 + 989*x^4 + 880*x^2 + 256);
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 + 24*x^12 + 110*x^10 + 176*x^8 + 440*x^6 + 989*x^4 + 880*x^2 + 256);
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
$C_2^2:C_4$ (as 16T10):
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 16 |
| The 10 conjugacy class representatives for $C_2^2 : C_4$ |
| Character table for $C_2^2 : C_4$ |
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
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.4.2 | $x^{4} + 4 x^{3} + 4 x^{2} + 12$ | $2$ | $2$ | $4$ | $C_4$ | $[2]^{2}$ |
| 2.4.4.2 | $x^{4} + 4 x^{3} + 4 x^{2} + 12$ | $2$ | $2$ | $4$ | $C_4$ | $[2]^{2}$ | |
| 2.4.4.2 | $x^{4} + 4 x^{3} + 4 x^{2} + 12$ | $2$ | $2$ | $4$ | $C_4$ | $[2]^{2}$ | |
| 2.4.4.2 | $x^{4} + 4 x^{3} + 4 x^{2} + 12$ | $2$ | $2$ | $4$ | $C_4$ | $[2]^{2}$ | |
|
\(5\)
| 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 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
|
\(11\)
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 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 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |