Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 2*x^14 + 7*x^13 - 15*x^12 + 12*x^11 - 6*x^10 + 16*x^9 - 24*x^8 - 5*x^7 + 67*x^6 - 114*x^5 + 110*x^4 - 70*x^3 + 30*x^2 - 8*x + 1)
gp: K = bnfinit(y^16 - 3*y^15 + 2*y^14 + 7*y^13 - 15*y^12 + 12*y^11 - 6*y^10 + 16*y^9 - 24*y^8 - 5*y^7 + 67*y^6 - 114*y^5 + 110*y^4 - 70*y^3 + 30*y^2 - 8*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 + 2*x^14 + 7*x^13 - 15*x^12 + 12*x^11 - 6*x^10 + 16*x^9 - 24*x^8 - 5*x^7 + 67*x^6 - 114*x^5 + 110*x^4 - 70*x^3 + 30*x^2 - 8*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 3*x^15 + 2*x^14 + 7*x^13 - 15*x^12 + 12*x^11 - 6*x^10 + 16*x^9 - 24*x^8 - 5*x^7 + 67*x^6 - 114*x^5 + 110*x^4 - 70*x^3 + 30*x^2 - 8*x + 1)
\( x^{16} - 3 x^{15} + 2 x^{14} + 7 x^{13} - 15 x^{12} + 12 x^{11} - 6 x^{10} + 16 x^{9} - 24 x^{8} + \cdots + 1 \)
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: |
\(232322192503515625\)
\(\medspace = 5^{8}\cdot 7^{2}\cdot 29^{4}\cdot 131^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(12.17\) | 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: | $5^{1/2}7^{1/2}29^{1/2}131^{1/2}\approx 364.64366167534024$ | ||
| Ramified primes: |
\(5\), \(7\), \(29\), \(131\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7}a^{14}+\frac{1}{7}a^{13}+\frac{2}{7}a^{12}-\frac{3}{7}a^{11}+\frac{3}{7}a^{9}-\frac{1}{7}a^{8}+\frac{1}{7}a^{6}-\frac{1}{7}a^{5}+\frac{3}{7}a^{4}+\frac{2}{7}$, $\frac{1}{133}a^{15}-\frac{1}{133}a^{14}+\frac{1}{19}a^{12}-\frac{1}{133}a^{11}+\frac{10}{133}a^{10}+\frac{2}{19}a^{9}+\frac{44}{133}a^{8}+\frac{64}{133}a^{7}-\frac{10}{133}a^{6}+\frac{47}{133}a^{5}-\frac{20}{133}a^{4}-\frac{9}{19}a^{3}-\frac{9}{19}a^{2}+\frac{37}{133}a+\frac{66}{133}$
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: |
$a^{15}-3a^{14}+2a^{13}+7a^{12}-15a^{11}+12a^{10}-6a^{9}+16a^{8}-24a^{7}-5a^{6}+67a^{5}-114a^{4}+110a^{3}-70a^{2}+30a-7$, $a$, $\frac{10558}{133}a^{15}-\frac{24922}{133}a^{14}+38a^{13}+\frac{11052}{19}a^{12}-\frac{108845}{133}a^{11}+\frac{56237}{133}a^{10}-\frac{3793}{19}a^{9}+\frac{151603}{133}a^{8}-\frac{156203}{133}a^{7}-\frac{154258}{133}a^{6}+\frac{609675}{133}a^{5}-\frac{811655}{133}a^{4}+\frac{90912}{19}a^{3}-\frac{46458}{19}a^{2}+\frac{104164}{133}a-\frac{16052}{133}$, $\frac{10558}{133}a^{15}-\frac{24922}{133}a^{14}+38a^{13}+\frac{11052}{19}a^{12}-\frac{108845}{133}a^{11}+\frac{56237}{133}a^{10}-\frac{3793}{19}a^{9}+\frac{151603}{133}a^{8}-\frac{156203}{133}a^{7}-\frac{154258}{133}a^{6}+\frac{609675}{133}a^{5}-\frac{811655}{133}a^{4}+\frac{90912}{19}a^{3}-\frac{46458}{19}a^{2}+\frac{104164}{133}a-\frac{15919}{133}$, $\frac{6183}{133}a^{15}-\frac{14676}{133}a^{14}+\frac{169}{7}a^{13}+\frac{45181}{133}a^{12}-\frac{64361}{133}a^{11}+\frac{34166}{133}a^{10}-\frac{16323}{133}a^{9}+\frac{89158}{133}a^{8}-\frac{92531}{133}a^{7}-\frac{88411}{133}a^{6}+\frac{357880}{133}a^{5}-\frac{480841}{133}a^{4}+\frac{54553}{19}a^{3}-\frac{28287}{19}a^{2}+\frac{64915}{133}a-\frac{10435}{133}$, $\frac{271}{133}a^{15}-\frac{112}{19}a^{14}+\frac{15}{7}a^{13}+\frac{2201}{133}a^{12}-\frac{3653}{133}a^{11}+\frac{1646}{133}a^{10}-\frac{671}{133}a^{9}+\frac{4324}{133}a^{8}-\frac{5798}{133}a^{7}-\frac{4420}{133}a^{6}+\frac{18304}{133}a^{5}-\frac{25180}{133}a^{4}+\frac{2843}{19}a^{3}-\frac{1508}{19}a^{2}+\frac{3510}{133}a-\frac{563}{133}$, $66a^{15}-\frac{1088}{7}a^{14}+\frac{228}{7}a^{13}+\frac{3368}{7}a^{12}-\frac{4758}{7}a^{11}+360a^{10}-\frac{1199}{7}a^{9}+\frac{6632}{7}a^{8}-974a^{7}-\frac{6625}{7}a^{6}+\frac{26610}{7}a^{5}-\frac{35618}{7}a^{4}+4024a^{3}-2078a^{2}+675a-\frac{748}{7}$, $\frac{11667}{133}a^{15}-\frac{27969}{133}a^{14}+\frac{332}{7}a^{13}+\frac{85773}{133}a^{12}-\frac{123159}{133}a^{11}+\frac{64534}{133}a^{10}-\frac{30139}{133}a^{9}+\frac{168555}{133}a^{8}-\frac{178194}{133}a^{7}-\frac{24012}{19}a^{6}+\frac{681425}{133}a^{5}-\frac{915459}{133}a^{4}+\frac{103408}{19}a^{3}-\frac{53304}{19}a^{2}+\frac{121257}{133}a-\frac{19086}{133}$, $\frac{5184}{133}a^{15}-\frac{12385}{133}a^{14}+\frac{146}{7}a^{13}+\frac{38112}{133}a^{12}-\frac{54603}{133}a^{11}+\frac{28565}{133}a^{10}-\frac{12867}{133}a^{9}+\frac{74633}{133}a^{8}-\frac{78928}{133}a^{7}-\frac{74602}{133}a^{6}+\frac{303118}{133}a^{5}-\frac{405647}{133}a^{4}+\frac{45589}{19}a^{3}-\frac{23286}{19}a^{2}+\frac{52158}{133}a-\frac{8083}{133}$
| 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: | \( 155.818339721 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 155.818339721 \cdot 1}{2\cdot\sqrt{232322192503515625}}\cr\approx \mathstrut & 0.159126467141 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 2*x^14 + 7*x^13 - 15*x^12 + 12*x^11 - 6*x^10 + 16*x^9 - 24*x^8 - 5*x^7 + 67*x^6 - 114*x^5 + 110*x^4 - 70*x^3 + 30*x^2 - 8*x + 1)
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 - 3*x^15 + 2*x^14 + 7*x^13 - 15*x^12 + 12*x^11 - 6*x^10 + 16*x^9 - 24*x^8 - 5*x^7 + 67*x^6 - 114*x^5 + 110*x^4 - 70*x^3 + 30*x^2 - 8*x + 1, 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 - 3*x^15 + 2*x^14 + 7*x^13 - 15*x^12 + 12*x^11 - 6*x^10 + 16*x^9 - 24*x^8 - 5*x^7 + 67*x^6 - 114*x^5 + 110*x^4 - 70*x^3 + 30*x^2 - 8*x + 1);
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 - 3*x^15 + 2*x^14 + 7*x^13 - 15*x^12 + 12*x^11 - 6*x^10 + 16*x^9 - 24*x^8 - 5*x^7 + 67*x^6 - 114*x^5 + 110*x^4 - 70*x^3 + 30*x^2 - 8*x + 1);
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
$C_2^7.C_2\wr D_4$ (as 16T1772):
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 16384 |
| The 148 conjugacy class representatives for $C_2^7.C_2\wr D_4$ |
| Character table for $C_2^7.C_2\wr D_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.725.1, 8.6.68856875.1 |
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]
Sibling fields
| Degree 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | R | R | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.16.8.1 | $x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
|
\(7\)
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.4.0.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 7.4.2.2 | $x^{4} - 42 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
|
\(29\)
| 29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.8.4.1 | $x^{8} + 2784 x^{7} + 2906616 x^{6} + 1348864734 x^{5} + 234834277018 x^{4} + 41857830864 x^{3} + 492109772617 x^{2} + 3561769809750 x + 616658760166$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(131\)
| 131.4.2.1 | $x^{4} + 254 x^{3} + 16395 x^{2} + 33782 x + 2129540$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 131.4.0.1 | $x^{4} + 9 x^{2} + 109 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 131.8.0.1 | $x^{8} + 3 x^{4} + 72 x^{3} + 116 x^{2} + 104 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |