Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 48*x^14 - 196*x^13 + 656*x^12 - 1752*x^11 + 3898*x^10 - 7148*x^9 + 10467*x^8 - 11960*x^7 + 8262*x^6 - 264*x^5 + 372*x^4 - 6272*x^3 + 4376*x^2 - 480*x + 16)
gp: K = bnfinit(y^16 - 8*y^15 + 48*y^14 - 196*y^13 + 656*y^12 - 1752*y^11 + 3898*y^10 - 7148*y^9 + 10467*y^8 - 11960*y^7 + 8262*y^6 - 264*y^5 + 372*y^4 - 6272*y^3 + 4376*y^2 - 480*y + 16, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 48*x^14 - 196*x^13 + 656*x^12 - 1752*x^11 + 3898*x^10 - 7148*x^9 + 10467*x^8 - 11960*x^7 + 8262*x^6 - 264*x^5 + 372*x^4 - 6272*x^3 + 4376*x^2 - 480*x + 16);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 48*x^14 - 196*x^13 + 656*x^12 - 1752*x^11 + 3898*x^10 - 7148*x^9 + 10467*x^8 - 11960*x^7 + 8262*x^6 - 264*x^5 + 372*x^4 - 6272*x^3 + 4376*x^2 - 480*x + 16)
\( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 656 x^{12} - 1752 x^{11} + 3898 x^{10} - 7148 x^{9} + \cdots + 16 \)
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: |
\(23612624896000000000000\)
\(\medspace = 2^{24}\cdot 5^{12}\cdot 7^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(25.02\) | 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: | $2^{3/2}5^{3/4}7^{1/2}\approx 25.021971019484877$ | ||
| Ramified primes: |
\(2\), \(5\), \(7\)
| 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 and abelian over $\Q$. | |||
| Conductor: | \(280=2^{3}\cdot 5\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{280}(1,·)$, $\chi_{280}(69,·)$, $\chi_{280}(13,·)$, $\chi_{280}(141,·)$, $\chi_{280}(209,·)$, $\chi_{280}(153,·)$, $\chi_{280}(29,·)$, $\chi_{280}(197,·)$, $\chi_{280}(97,·)$, $\chi_{280}(41,·)$, $\chi_{280}(237,·)$, $\chi_{280}(113,·)$, $\chi_{280}(181,·)$, $\chi_{280}(169,·)$, $\chi_{280}(57,·)$, $\chi_{280}(253,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{5}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{6}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{7}$, $\frac{1}{40}a^{12}+\frac{1}{10}a^{11}-\frac{3}{40}a^{10}-\frac{1}{40}a^{8}+\frac{1}{5}a^{7}-\frac{9}{40}a^{6}+\frac{1}{10}a^{5}-\frac{1}{10}a^{4}+\frac{2}{5}a^{3}+\frac{2}{5}a^{2}+\frac{1}{5}a+\frac{2}{5}$, $\frac{1}{40}a^{13}+\frac{1}{40}a^{11}+\frac{1}{20}a^{10}-\frac{1}{40}a^{9}+\frac{1}{20}a^{8}-\frac{1}{40}a^{7}-\frac{1}{4}a^{6}+\frac{1}{20}a^{4}-\frac{1}{5}a^{3}+\frac{1}{10}a^{2}-\frac{2}{5}a+\frac{2}{5}$, $\frac{1}{847507160}a^{14}-\frac{7}{847507160}a^{13}+\frac{8689041}{847507160}a^{12}-\frac{10426831}{169501432}a^{11}+\frac{40085}{936472}a^{10}+\frac{84634839}{847507160}a^{9}-\frac{456891}{8921128}a^{8}-\frac{60158713}{847507160}a^{7}-\frac{214483}{1389356}a^{6}-\frac{15079731}{105938395}a^{5}-\frac{35775893}{211876790}a^{4}+\frac{19241563}{42375358}a^{3}-\frac{104233}{585295}a^{2}+\frac{23346361}{105938395}a+\frac{7502711}{105938395}$, $\frac{1}{2801011163800}a^{15}+\frac{329}{560202232760}a^{14}-\frac{2336899791}{1400505581900}a^{13}+\frac{9725622913}{2801011163800}a^{12}-\frac{8407651763}{280101116380}a^{11}+\frac{274560552493}{2801011163800}a^{10}+\frac{92991838641}{1400505581900}a^{9}-\frac{214696714787}{2801011163800}a^{8}-\frac{343319650759}{2801011163800}a^{7}-\frac{45265879711}{1400505581900}a^{6}-\frac{65514602683}{350126395475}a^{5}-\frac{44892454613}{1400505581900}a^{4}-\frac{83081896592}{350126395475}a^{3}-\frac{29662287261}{70025279095}a^{2}-\frac{127620462993}{350126395475}a-\frac{173188401789}{350126395475}$
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: |
\( \frac{3162903}{423753580} a^{14} - \frac{22140321}{423753580} a^{13} + \frac{259143999}{847507160} a^{12} - \frac{122401956}{105938395} a^{11} + \frac{17468601}{4682360} a^{10} - \frac{3944147883}{423753580} a^{9} + \frac{877017393}{44605640} a^{8} - \frac{7071434721}{211876790} a^{7} + \frac{612123373}{13893560} a^{6} - \frac{9319400199}{211876790} a^{5} + \frac{6724200103}{423753580} a^{4} + \frac{1697853116}{105938395} a^{3} + \frac{20065621}{1170590} a^{2} - \frac{3064218226}{105938395} a + \frac{272112387}{105938395} \)
(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: |
$\frac{5109}{22302820}a^{14}-\frac{35763}{22302820}a^{13}+\frac{439941}{44605640}a^{12}-\frac{213726}{5575705}a^{11}+\frac{32411}{246440}a^{10}-\frac{7681709}{22302820}a^{9}+\frac{35324717}{44605640}a^{8}-\frac{3250195}{2230282}a^{7}+\frac{1638423}{731240}a^{6}-\frac{30276173}{11151410}a^{5}+\frac{47234981}{22302820}a^{4}-\frac{5008004}{5575705}a^{3}+\frac{5437}{30805}a^{2}-\frac{76304}{5575705}a+\frac{3448094}{5575705}$, $\frac{7670780656}{350126395475}a^{15}-\frac{43934005053}{280101116380}a^{14}+\frac{1291140434127}{1400505581900}a^{13}-\frac{9899203221761}{2801011163800}a^{12}+\frac{804547362321}{70025279095}a^{11}-\frac{81394795041231}{2801011163800}a^{10}+\frac{86901651643683}{1400505581900}a^{9}-\frac{300087714381561}{2801011163800}a^{8}+\frac{101513931030517}{700252790950}a^{7}-\frac{419927826428551}{2801011163800}a^{6}+\frac{47670228117517}{700252790950}a^{5}+\frac{53012805869191}{1400505581900}a^{4}+\frac{32401596146493}{700252790950}a^{3}-\frac{13506081567357}{140050558190}a^{2}+\frac{7909110382356}{350126395475}a-\frac{500659210857}{350126395475}$, $\frac{34585042681}{1400505581900}a^{15}-\frac{97270321443}{560202232760}a^{14}+\frac{2845353972881}{2801011163800}a^{13}-\frac{10775156559419}{2801011163800}a^{12}+\frac{6957488262053}{560202232760}a^{11}-\frac{74996338571}{2416748200}a^{10}+\frac{183609934207259}{2801011163800}a^{9}-\frac{312060588661339}{2801011163800}a^{8}+\frac{412314565807817}{2801011163800}a^{7}-\frac{103026772508821}{700252790950}a^{6}+\frac{19013786243249}{350126395475}a^{5}+\frac{74107740950719}{1400505581900}a^{4}+\frac{1086125360229}{18427705025}a^{3}-\frac{7136860688489}{70025279095}a^{2}+\frac{2346218004074}{350126395475}a-\frac{84738305453}{350126395475}$, $\frac{10203924346}{350126395475}a^{15}-\frac{56903721369}{280101116380}a^{14}+\frac{1663006752747}{1400505581900}a^{13}-\frac{12530147810761}{2801011163800}a^{12}+\frac{201909014611}{14005055819}a^{11}-\frac{100426764929151}{2801011163800}a^{10}+\frac{105752804485853}{1400505581900}a^{9}-\frac{3539419844431}{27732783800}a^{8}+\frac{117210834203127}{700252790950}a^{7}-\frac{463641704752491}{2801011163800}a^{6}+\frac{39288611951517}{700252790950}a^{5}+\frac{22602115892494}{350126395475}a^{4}+\frac{25103010526934}{350126395475}a^{3}-\frac{419475263736}{3685541005}a^{2}+\frac{1947214113716}{350126395475}a+\frac{56338474073}{350126395475}$, $\frac{2905214158}{350126395475}a^{15}-\frac{17292818007}{280101116380}a^{14}+\frac{511403129541}{1400505581900}a^{13}-\frac{4011551030023}{2801011163800}a^{12}+\frac{329214021541}{70025279095}a^{11}-\frac{33989064594473}{2801011163800}a^{10}+\frac{36819409162759}{1400505581900}a^{9}-\frac{130109002208453}{2801011163800}a^{8}+\frac{45349256390301}{700252790950}a^{7}-\frac{194979353015413}{2801011163800}a^{6}+\frac{26749581447771}{700252790950}a^{5}+\frac{7774633034529}{700252790950}a^{4}+\frac{3852988023222}{350126395475}a^{3}-\frac{5550747796591}{140050558190}a^{2}+\frac{4600633708958}{350126395475}a-\frac{44643912646}{350126395475}$, $\frac{681618159}{36855410050}a^{15}-\frac{72371387943}{560202232760}a^{14}+\frac{264385428889}{350126395475}a^{13}-\frac{3988536889009}{1400505581900}a^{12}+\frac{1285620913979}{140050558190}a^{11}-\frac{32000953719199}{1400505581900}a^{10}+\frac{67418556390999}{1400505581900}a^{9}-\frac{6002877408571}{73710820100}a^{8}+\frac{1227283262871}{11479553950}a^{7}-\frac{296527406009763}{2801011163800}a^{6}+\frac{50869756815037}{1400505581900}a^{5}+\frac{14369701317742}{350126395475}a^{4}+\frac{31671212906739}{700252790950}a^{3}-\frac{5152785559402}{70025279095}a^{2}+\frac{1126005458758}{350126395475}a+\frac{370604}{56829475}$, $\frac{5315520867}{1400505581900}a^{15}-\frac{17162450949}{560202232760}a^{14}+\frac{514879877177}{2801011163800}a^{13}-\frac{1054910736189}{1400505581900}a^{12}+\frac{1412541077803}{560202232760}a^{11}-\frac{2362390005481}{350126395475}a^{10}+\frac{42062499904223}{2801011163800}a^{9}-\frac{19307821152027}{700252790950}a^{8}+\frac{113262370836419}{2801011163800}a^{7}-\frac{129611057055043}{2801011163800}a^{6}+\frac{22704753903311}{700252790950}a^{5}-\frac{124340559123}{73710820100}a^{4}+\frac{1863433189059}{700252790950}a^{3}-\frac{3820696169421}{140050558190}a^{2}+\frac{6188570288363}{350126395475}a-\frac{204587998211}{350126395475}$
| 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: | \( 34736.104354 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 34736.104354 \cdot 4}{10\cdot\sqrt{23612624896000000000000}}\cr\approx \mathstrut & 0.21963821977 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 48*x^14 - 196*x^13 + 656*x^12 - 1752*x^11 + 3898*x^10 - 7148*x^9 + 10467*x^8 - 11960*x^7 + 8262*x^6 - 264*x^5 + 372*x^4 - 6272*x^3 + 4376*x^2 - 480*x + 16)
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 - 8*x^15 + 48*x^14 - 196*x^13 + 656*x^12 - 1752*x^11 + 3898*x^10 - 7148*x^9 + 10467*x^8 - 11960*x^7 + 8262*x^6 - 264*x^5 + 372*x^4 - 6272*x^3 + 4376*x^2 - 480*x + 16, 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 - 8*x^15 + 48*x^14 - 196*x^13 + 656*x^12 - 1752*x^11 + 3898*x^10 - 7148*x^9 + 10467*x^8 - 11960*x^7 + 8262*x^6 - 264*x^5 + 372*x^4 - 6272*x^3 + 4376*x^2 - 480*x + 16);
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 - 8*x^15 + 48*x^14 - 196*x^13 + 656*x^12 - 1752*x^11 + 3898*x^10 - 7148*x^9 + 10467*x^8 - 11960*x^7 + 8262*x^6 - 264*x^5 + 372*x^4 - 6272*x^3 + 4376*x^2 - 480*x + 16);
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
$C_2^2\times C_4$ (as 16T2):
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)
| An abelian group of order 16 |
| The 16 conjugacy class representatives for $C_4\times C_2^2$ |
| Character table for $C_4\times C_2^2$ |
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]
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 | R | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.12.1 | $x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ |
| 2.8.12.1 | $x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
|
\(5\)
| 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 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
|
\(7\)
| 7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |