Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 15*x^14 - 32*x^13 + 56*x^12 - 85*x^11 + 114*x^10 - 134*x^9 + 131*x^8 - 98*x^7 + 42*x^6 + 9*x^5 - 28*x^4 + 18*x^3 - x^2 - 3*x + 1)
gp: K = bnfinit(y^16 - 5*y^15 + 15*y^14 - 32*y^13 + 56*y^12 - 85*y^11 + 114*y^10 - 134*y^9 + 131*y^8 - 98*y^7 + 42*y^6 + 9*y^5 - 28*y^4 + 18*y^3 - y^2 - 3*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 5*x^15 + 15*x^14 - 32*x^13 + 56*x^12 - 85*x^11 + 114*x^10 - 134*x^9 + 131*x^8 - 98*x^7 + 42*x^6 + 9*x^5 - 28*x^4 + 18*x^3 - x^2 - 3*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 5*x^15 + 15*x^14 - 32*x^13 + 56*x^12 - 85*x^11 + 114*x^10 - 134*x^9 + 131*x^8 - 98*x^7 + 42*x^6 + 9*x^5 - 28*x^4 + 18*x^3 - x^2 - 3*x + 1)
\( x^{16} - 5 x^{15} + 15 x^{14} - 32 x^{13} + 56 x^{12} - 85 x^{11} + 114 x^{10} - 134 x^{9} + 131 x^{8} + \cdots + 1 \)
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: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(6561000000000000\)
\(\medspace = 2^{12}\cdot 3^{8}\cdot 5^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(9.74\) | 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: | $2^{3/2}3^{1/2}5^{3/4}\approx 16.3807251762544$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{13129}a^{15}-\frac{1370}{13129}a^{14}+\frac{5747}{13129}a^{13}+\frac{6455}{13129}a^{12}-\frac{1460}{13129}a^{11}-\frac{147}{691}a^{10}+\frac{271}{691}a^{9}-\frac{4504}{13129}a^{8}+\frac{3719}{13129}a^{7}+\frac{4390}{13129}a^{6}-\frac{5484}{13129}a^{5}+\frac{2139}{13129}a^{4}-\frac{5125}{13129}a^{3}-\frac{2114}{13129}a^{2}-\frac{2771}{13129}a+\frac{1260}{13129}$
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: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{4711}{13129} a^{15} - \frac{20860}{13129} a^{14} + \frac{67764}{13129} a^{13} - \frac{154807}{13129} a^{12} + \frac{290374}{13129} a^{11} - \frac{24320}{691} a^{10} + \frac{34263}{691} a^{9} - \frac{815878}{13129} a^{8} + \frac{872637}{13129} a^{7} - \frac{771496}{13129} a^{6} + \frac{501650}{13129} a^{5} - \frac{163791}{13129} a^{4} - \frac{65289}{13129} a^{3} + \frac{110889}{13129} a^{2} - \frac{30213}{13129} a - \frac{11577}{13129} \)
(order $30$)
| 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: |
$\frac{458}{13129}a^{15}+\frac{2732}{13129}a^{14}-\frac{6803}{13129}a^{13}+\frac{15494}{13129}a^{12}-\frac{25359}{13129}a^{11}+\frac{2465}{691}a^{10}-\frac{3717}{691}a^{9}+\frac{90324}{13129}a^{8}-\frac{108500}{13129}a^{7}+\frac{106915}{13129}a^{6}-\frac{95936}{13129}a^{5}+\frac{86890}{13129}a^{4}-\frac{62804}{13129}a^{3}+\frac{29592}{13129}a^{2}+\frac{30653}{13129}a-\frac{13725}{13129}$, $\frac{9829}{13129}a^{15}-\frac{34763}{13129}a^{14}+\frac{85079}{13129}a^{13}-\frac{150681}{13129}a^{12}+\frac{235979}{13129}a^{11}-\frac{17257}{691}a^{10}+\frac{20584}{691}a^{9}-\frac{405827}{13129}a^{8}+\frac{304882}{13129}a^{7}-\frac{137003}{13129}a^{6}-\frac{33949}{13129}a^{5}+\frac{83476}{13129}a^{4}-\frac{10781}{13129}a^{3}-\frac{21557}{13129}a^{2}+\frac{32774}{13129}a+\frac{3893}{13129}$, $\frac{9392}{13129}a^{15}-\frac{40007}{13129}a^{14}+\frac{107537}{13129}a^{13}-\frac{214426}{13129}a^{12}+\frac{361968}{13129}a^{11}-\frac{28337}{691}a^{10}+\frac{36902}{691}a^{9}-\frac{800799}{13129}a^{8}+\frac{754061}{13129}a^{7}-\frac{532469}{13129}a^{6}+\frac{222532}{13129}a^{5}+\frac{28376}{13129}a^{4}-\frac{94989}{13129}a^{3}+\frac{62005}{13129}a^{2}-\frac{3554}{13129}a-\frac{8438}{13129}$, $\frac{12204}{13129}a^{15}-\frac{58779}{13129}a^{14}+\frac{171947}{13129}a^{13}-\frac{364792}{13129}a^{12}+\frac{641534}{13129}a^{11}-\frac{51977}{691}a^{10}+\frac{70640}{691}a^{9}-\frac{1610560}{13129}a^{8}+\frac{1627719}{13129}a^{7}-\frac{1316789}{13129}a^{6}+\frac{740130}{13129}a^{5}-\frac{179902}{13129}a^{4}-\frac{90847}{13129}a^{3}+\frac{104261}{13129}a^{2}-\frac{23238}{13129}a+\frac{2981}{13129}$, $\frac{2452}{13129}a^{15}-\frac{11345}{13129}a^{14}+\frac{30485}{13129}a^{13}-\frac{58430}{13129}a^{12}+\frac{96200}{13129}a^{11}-\frac{7343}{691}a^{10}+\frac{9424}{691}a^{9}-\frac{199254}{13129}a^{8}+\frac{178139}{13129}a^{7}-\frac{119661}{13129}a^{6}+\frac{36715}{13129}a^{5}+\frac{6357}{13129}a^{4}-\frac{2047}{13129}a^{3}-\frac{23831}{13129}a^{2}+\frac{19459}{13129}a+\frac{4205}{13129}$, $\frac{5442}{13129}a^{15}-\frac{24526}{13129}a^{14}+\frac{67541}{13129}a^{13}-\frac{136384}{13129}a^{12}+\frac{234047}{13129}a^{11}-\frac{18453}{691}a^{10}+\frac{24373}{691}a^{9}-\frac{537214}{13129}a^{8}+\frac{519040}{13129}a^{7}-\frac{385141}{13129}a^{6}+\frac{182095}{13129}a^{5}-\frac{18114}{13129}a^{4}-\frac{43641}{13129}a^{3}+\frac{36003}{13129}a^{2}+\frac{5439}{13129}a-\frac{9547}{13129}$, $\frac{5448}{13129}a^{15}-\frac{19617}{13129}a^{14}+\frac{49507}{13129}a^{13}-\frac{84525}{13129}a^{12}+\frac{120255}{13129}a^{11}-\frac{7588}{691}a^{10}+\frac{7342}{691}a^{9}-\frac{91594}{13129}a^{8}-\frac{36322}{13129}a^{7}+\frac{192617}{13129}a^{6}-\frac{323453}{13129}a^{5}+\frac{322945}{13129}a^{4}-\frac{166294}{13129}a^{3}+\frac{10190}{13129}a^{2}+\frac{67587}{13129}a-\frac{28245}{13129}$
| 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: | \( 175.01473438 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 175.01473438 \cdot 1}{30\cdot\sqrt{6561000000000000}}\cr\approx \mathstrut & 0.17494731698 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 15*x^14 - 32*x^13 + 56*x^12 - 85*x^11 + 114*x^10 - 134*x^9 + 131*x^8 - 98*x^7 + 42*x^6 + 9*x^5 - 28*x^4 + 18*x^3 - x^2 - 3*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 - 5*x^15 + 15*x^14 - 32*x^13 + 56*x^12 - 85*x^11 + 114*x^10 - 134*x^9 + 131*x^8 - 98*x^7 + 42*x^6 + 9*x^5 - 28*x^4 + 18*x^3 - x^2 - 3*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 - 5*x^15 + 15*x^14 - 32*x^13 + 56*x^12 - 85*x^11 + 114*x^10 - 134*x^9 + 131*x^8 - 98*x^7 + 42*x^6 + 9*x^5 - 28*x^4 + 18*x^3 - x^2 - 3*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 - 5*x^15 + 15*x^14 - 32*x^13 + 56*x^12 - 85*x^11 + 114*x^10 - 134*x^9 + 131*x^8 - 98*x^7 + 42*x^6 + 9*x^5 - 28*x^4 + 18*x^3 - x^2 - 3*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
$C_4\times D_4$ (as 16T19):
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 $C_4 \times D_4$ |
| Character table for $C_4 \times D_4$ |
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: | 16.0.331776000000000000.1, 16.8.26873856000000000000.4, 16.0.26873856000000000000.11 |
| 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 | R | R | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 2.8.12.1 | $x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
|
\(3\)
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(5\)
| 5.16.12.1 | $x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$ | $4$ | $4$ | $12$ | $C_4^2$ | $[\ ]_{4}^{4}$ |