Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 488*x^14 + 71756*x^12 + 2873344*x^10 + 14583259*x^8 - 687937848*x^6 + 3865158504*x^4 - 409240944*x^2 + 89663714721)
gp: K = bnfinit(y^16 + 488*y^14 + 71756*y^12 + 2873344*y^10 + 14583259*y^8 - 687937848*y^6 + 3865158504*y^4 - 409240944*y^2 + 89663714721, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 488*x^14 + 71756*x^12 + 2873344*x^10 + 14583259*x^8 - 687937848*x^6 + 3865158504*x^4 - 409240944*x^2 + 89663714721);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 488*x^14 + 71756*x^12 + 2873344*x^10 + 14583259*x^8 - 687937848*x^6 + 3865158504*x^4 - 409240944*x^2 + 89663714721)
\( x^{16} + 488 x^{14} + 71756 x^{12} + 2873344 x^{10} + 14583259 x^{8} - 687937848 x^{6} + \cdots + 89663714721 \)
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: |
\(262686006513622818702917369856000000000000\)
\(\medspace = 2^{48}\cdot 3^{8}\cdot 5^{12}\cdot 17^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(387.90\) | 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: | $2^{3}3^{1/2}5^{3/4}17^{3/4}\approx 387.89556009707036$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(17\)
| 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: | \(4080=2^{4}\cdot 3\cdot 5\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{4080}(1,·)$, $\chi_{4080}(2821,·)$, $\chi_{4080}(781,·)$, $\chi_{4080}(3469,·)$, $\chi_{4080}(3107,·)$, $\chi_{4080}(1429,·)$, $\chi_{4080}(2903,·)$, $\chi_{4080}(2843,·)$, $\chi_{4080}(863,·)$, $\chi_{4080}(2209,·)$, $\chi_{4080}(803,·)$, $\chi_{4080}(3047,·)$, $\chi_{4080}(169,·)$, $\chi_{4080}(1067,·)$, $\chi_{4080}(1007,·)$, $\chi_{4080}(2041,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{7}a^{5}+\frac{1}{7}a^{3}+\frac{1}{7}a$, $\frac{1}{7}a^{6}+\frac{1}{7}a^{4}+\frac{1}{7}a^{2}$, $\frac{1}{7}a^{7}-\frac{1}{7}a$, $\frac{1}{7}a^{8}-\frac{1}{7}a^{2}$, $\frac{1}{7}a^{9}-\frac{1}{7}a^{3}$, $\frac{1}{147}a^{10}+\frac{2}{147}a^{8}-\frac{4}{147}a^{6}-\frac{5}{147}a^{4}+\frac{43}{147}a^{2}$, $\frac{1}{147}a^{11}+\frac{2}{147}a^{9}-\frac{4}{147}a^{7}-\frac{5}{147}a^{5}+\frac{43}{147}a^{3}$, $\frac{1}{122157}a^{12}-\frac{130}{122157}a^{10}-\frac{4090}{122157}a^{8}+\frac{4807}{122157}a^{6}+\frac{59377}{122157}a^{4}-\frac{3061}{40719}a^{2}+\frac{60}{277}$, $\frac{1}{1741836663}a^{13}-\frac{3580}{248833809}a^{11}-\frac{16963969}{1741836663}a^{9}+\frac{24780241}{1741836663}a^{7}+\frac{403630}{17957079}a^{5}+\frac{21686039}{580612221}a^{3}-\frac{118219}{3949743}a$, $\frac{1}{12\!\cdots\!41}a^{14}-\frac{96\!\cdots\!73}{17\!\cdots\!63}a^{12}-\frac{21\!\cdots\!10}{12\!\cdots\!41}a^{10}-\frac{34\!\cdots\!20}{12\!\cdots\!41}a^{8}-\frac{32\!\cdots\!35}{12\!\cdots\!53}a^{6}+\frac{40\!\cdots\!95}{45\!\cdots\!83}a^{4}+\frac{60\!\cdots\!10}{27\!\cdots\!01}a^{2}-\frac{52\!\cdots\!06}{19\!\cdots\!39}$, $\frac{1}{36\!\cdots\!23}a^{15}-\frac{77\!\cdots\!70}{36\!\cdots\!23}a^{13}-\frac{24\!\cdots\!64}{36\!\cdots\!23}a^{11}+\frac{33\!\cdots\!12}{52\!\cdots\!89}a^{9}+\frac{59\!\cdots\!77}{36\!\cdots\!23}a^{7}-\frac{63\!\cdots\!59}{40\!\cdots\!47}a^{5}+\frac{20\!\cdots\!56}{40\!\cdots\!47}a^{3}-\frac{21\!\cdots\!49}{27\!\cdots\!01}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $7$ |
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{20}\times C_{120}\times C_{6960}$, which has order $267264000$ (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: |
\( -1 \)
(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: |
$\frac{21564408260}{47\!\cdots\!29}a^{14}+\frac{1524307832290}{67\!\cdots\!47}a^{12}+\frac{16\!\cdots\!07}{47\!\cdots\!29}a^{10}+\frac{72\!\cdots\!50}{47\!\cdots\!29}a^{8}+\frac{70\!\cdots\!35}{48\!\cdots\!57}a^{6}-\frac{54\!\cdots\!10}{17\!\cdots\!27}a^{4}-\frac{22\!\cdots\!95}{35\!\cdots\!23}a^{2}-\frac{13698669443423}{75036723895291}$, $\frac{45\!\cdots\!60}{20\!\cdots\!97}a^{15}+\frac{43128816520}{47\!\cdots\!29}a^{14}+\frac{21\!\cdots\!67}{20\!\cdots\!97}a^{13}+\frac{3048615664580}{67\!\cdots\!47}a^{12}+\frac{97\!\cdots\!82}{68\!\cdots\!99}a^{11}+\frac{32\!\cdots\!14}{47\!\cdots\!29}a^{10}+\frac{76\!\cdots\!59}{20\!\cdots\!97}a^{9}+\frac{14\!\cdots\!00}{47\!\cdots\!29}a^{8}-\frac{50\!\cdots\!36}{68\!\cdots\!99}a^{7}+\frac{14\!\cdots\!70}{48\!\cdots\!57}a^{6}-\frac{42\!\cdots\!66}{20\!\cdots\!97}a^{5}-\frac{10\!\cdots\!20}{17\!\cdots\!27}a^{4}+\frac{24\!\cdots\!67}{68\!\cdots\!99}a^{3}-\frac{44\!\cdots\!90}{35\!\cdots\!23}a^{2}-\frac{50\!\cdots\!97}{46\!\cdots\!17}a+\frac{197712832799027}{75036723895291}$, $\frac{35\!\cdots\!35}{87\!\cdots\!13}a^{15}+\frac{43128816520}{47\!\cdots\!29}a^{14}+\frac{12\!\cdots\!36}{61\!\cdots\!91}a^{13}+\frac{3048615664580}{67\!\cdots\!47}a^{12}+\frac{25\!\cdots\!78}{87\!\cdots\!13}a^{11}+\frac{32\!\cdots\!14}{47\!\cdots\!29}a^{10}+\frac{73\!\cdots\!92}{61\!\cdots\!91}a^{9}+\frac{14\!\cdots\!00}{47\!\cdots\!29}a^{8}+\frac{47\!\cdots\!48}{61\!\cdots\!91}a^{7}+\frac{14\!\cdots\!70}{48\!\cdots\!57}a^{6}-\frac{52\!\cdots\!79}{20\!\cdots\!97}a^{5}-\frac{10\!\cdots\!20}{17\!\cdots\!27}a^{4}+\frac{79\!\cdots\!94}{68\!\cdots\!99}a^{3}-\frac{44\!\cdots\!90}{35\!\cdots\!23}a^{2}-\frac{12\!\cdots\!61}{15\!\cdots\!39}a+\frac{47639385008445}{75036723895291}$, $\frac{18484455262136}{24\!\cdots\!01}a^{14}+\frac{12\!\cdots\!32}{34\!\cdots\!43}a^{12}+\frac{10\!\cdots\!64}{24\!\cdots\!01}a^{10}+\frac{21\!\cdots\!21}{24\!\cdots\!01}a^{8}-\frac{45\!\cdots\!04}{24\!\cdots\!01}a^{6}+\frac{22\!\cdots\!68}{24\!\cdots\!01}a^{4}-\frac{24\!\cdots\!64}{34\!\cdots\!43}a^{2}-\frac{62\!\cdots\!02}{70\!\cdots\!07}$, $\frac{24\!\cdots\!13}{36\!\cdots\!23}a^{15}+\frac{20\!\cdots\!76}{12\!\cdots\!41}a^{14}+\frac{11\!\cdots\!16}{36\!\cdots\!23}a^{13}+\frac{14\!\cdots\!14}{17\!\cdots\!63}a^{12}+\frac{18\!\cdots\!25}{36\!\cdots\!23}a^{11}+\frac{15\!\cdots\!57}{12\!\cdots\!41}a^{10}+\frac{80\!\cdots\!67}{36\!\cdots\!23}a^{9}+\frac{67\!\cdots\!31}{12\!\cdots\!41}a^{8}+\frac{80\!\cdots\!19}{36\!\cdots\!23}a^{7}+\frac{62\!\cdots\!49}{12\!\cdots\!53}a^{6}-\frac{14\!\cdots\!73}{40\!\cdots\!47}a^{5}-\frac{49\!\cdots\!02}{45\!\cdots\!83}a^{4}-\frac{20\!\cdots\!50}{40\!\cdots\!47}a^{3}-\frac{20\!\cdots\!49}{92\!\cdots\!67}a^{2}-\frac{21\!\cdots\!01}{27\!\cdots\!01}a-\frac{37\!\cdots\!54}{19\!\cdots\!39}$, $\frac{62\!\cdots\!88}{36\!\cdots\!23}a^{15}-\frac{55\!\cdots\!34}{12\!\cdots\!41}a^{14}+\frac{30\!\cdots\!25}{36\!\cdots\!23}a^{13}-\frac{38\!\cdots\!54}{17\!\cdots\!63}a^{12}+\frac{46\!\cdots\!23}{36\!\cdots\!23}a^{11}-\frac{40\!\cdots\!67}{12\!\cdots\!41}a^{10}+\frac{20\!\cdots\!90}{36\!\cdots\!23}a^{9}-\frac{17\!\cdots\!99}{12\!\cdots\!41}a^{8}+\frac{21\!\cdots\!97}{36\!\cdots\!23}a^{7}-\frac{16\!\cdots\!15}{12\!\cdots\!53}a^{6}-\frac{11\!\cdots\!90}{12\!\cdots\!41}a^{5}+\frac{12\!\cdots\!02}{45\!\cdots\!83}a^{4}-\frac{25\!\cdots\!31}{13\!\cdots\!49}a^{3}+\frac{53\!\cdots\!33}{92\!\cdots\!67}a^{2}-\frac{51\!\cdots\!51}{27\!\cdots\!01}a+\frac{10\!\cdots\!29}{19\!\cdots\!39}$, $\frac{19\!\cdots\!19}{36\!\cdots\!23}a^{15}-\frac{53\!\cdots\!82}{12\!\cdots\!41}a^{14}+\frac{14\!\cdots\!04}{52\!\cdots\!89}a^{13}-\frac{37\!\cdots\!30}{17\!\cdots\!63}a^{12}+\frac{16\!\cdots\!91}{36\!\cdots\!23}a^{11}-\frac{39\!\cdots\!19}{12\!\cdots\!41}a^{10}+\frac{94\!\cdots\!21}{36\!\cdots\!23}a^{9}-\frac{17\!\cdots\!77}{12\!\cdots\!41}a^{8}+\frac{18\!\cdots\!51}{36\!\cdots\!23}a^{7}-\frac{16\!\cdots\!39}{12\!\cdots\!53}a^{6}-\frac{13\!\cdots\!31}{12\!\cdots\!41}a^{5}+\frac{12\!\cdots\!90}{45\!\cdots\!83}a^{4}-\frac{58\!\cdots\!52}{64\!\cdots\!69}a^{3}+\frac{53\!\cdots\!01}{92\!\cdots\!67}a^{2}+\frac{51\!\cdots\!15}{39\!\cdots\!43}a-\frac{71\!\cdots\!30}{19\!\cdots\!39}$
| 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: | \( 6183243.81077213 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 6183243.81077213 \cdot 267264000}{2\cdot\sqrt{262686006513622818702917369856000000000000}}\cr\approx \mathstrut & 3.91604178825201 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 + 488*x^14 + 71756*x^12 + 2873344*x^10 + 14583259*x^8 - 687937848*x^6 + 3865158504*x^4 - 409240944*x^2 + 89663714721)
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 + 488*x^14 + 71756*x^12 + 2873344*x^10 + 14583259*x^8 - 687937848*x^6 + 3865158504*x^4 - 409240944*x^2 + 89663714721, 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 + 488*x^14 + 71756*x^12 + 2873344*x^10 + 14583259*x^8 - 687937848*x^6 + 3865158504*x^4 - 409240944*x^2 + 89663714721);
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 + 488*x^14 + 71756*x^12 + 2873344*x^10 + 14583259*x^8 - 687937848*x^6 + 3865158504*x^4 - 409240944*x^2 + 89663714721);
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
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 $C_4^2$ |
| Character table for $C_4^2$ |
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 | R | R | ${\href{/padicField/7.1.0.1}{1} }^{16}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.1.0.1}{1} }^{16}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $16$ | $8$ | $2$ | $48$ | |||
|
\(3\)
| 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 | $x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$ | $4$ | $4$ | $12$ | $C_4^2$ | $[\ ]_{4}^{4}$ |
|
\(17\)
| 17.8.6.2 | $x^{8} + 204 x^{4} - 7225$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 17.8.6.2 | $x^{8} + 204 x^{4} - 7225$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |