Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 4*x^14 + 9*x^13 - 16*x^12 - 16*x^11 + 23*x^10 + 140*x^9 - 222*x^8 - 317*x^7 + 1076*x^6 - 632*x^5 - 455*x^4 + 462*x^3 + 36*x^2 - 88*x + 16)
gp: K = bnfinit(y^16 - 4*y^15 + 4*y^14 + 9*y^13 - 16*y^12 - 16*y^11 + 23*y^10 + 140*y^9 - 222*y^8 - 317*y^7 + 1076*y^6 - 632*y^5 - 455*y^4 + 462*y^3 + 36*y^2 - 88*y + 16, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 4*x^14 + 9*x^13 - 16*x^12 - 16*x^11 + 23*x^10 + 140*x^9 - 222*x^8 - 317*x^7 + 1076*x^6 - 632*x^5 - 455*x^4 + 462*x^3 + 36*x^2 - 88*x + 16);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 + 4*x^14 + 9*x^13 - 16*x^12 - 16*x^11 + 23*x^10 + 140*x^9 - 222*x^8 - 317*x^7 + 1076*x^6 - 632*x^5 - 455*x^4 + 462*x^3 + 36*x^2 - 88*x + 16)
\( x^{16} - 4 x^{15} + 4 x^{14} + 9 x^{13} - 16 x^{12} - 16 x^{11} + 23 x^{10} + 140 x^{9} - 222 x^{8} + \cdots + 16 \)
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: |
\(1644508157944864993729\)
\(\medspace = 13^{8}\cdot 17^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(21.18\) | 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: | $13^{1/2}17^{3/4}\approx 30.186194580660594$ | ||
| Ramified primes: |
\(13\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{6}a^{13}+\frac{1}{3}a^{12}-\frac{1}{6}a^{10}-\frac{1}{3}a^{9}-\frac{1}{2}a^{7}+\frac{1}{6}a^{4}+\frac{1}{3}a^{3}-\frac{1}{2}a+\frac{1}{3}$, $\frac{1}{12}a^{14}-\frac{1}{3}a^{12}+\frac{5}{12}a^{11}+\frac{1}{3}a^{9}+\frac{1}{4}a^{8}-\frac{1}{2}a^{6}-\frac{5}{12}a^{5}-\frac{1}{3}a^{3}-\frac{1}{4}a^{2}+\frac{1}{6}a-\frac{1}{3}$, $\frac{1}{14\!\cdots\!36}a^{15}-\frac{16\!\cdots\!65}{70\!\cdots\!68}a^{14}+\frac{16\!\cdots\!97}{35\!\cdots\!34}a^{13}-\frac{27\!\cdots\!67}{14\!\cdots\!36}a^{12}+\frac{22\!\cdots\!45}{70\!\cdots\!68}a^{11}+\frac{66\!\cdots\!62}{17\!\cdots\!67}a^{10}-\frac{26\!\cdots\!07}{46\!\cdots\!12}a^{9}+\frac{93497679251305}{543361206164092}a^{8}-\frac{44\!\cdots\!81}{23\!\cdots\!56}a^{7}-\frac{66\!\cdots\!13}{14\!\cdots\!36}a^{6}+\frac{23\!\cdots\!81}{70\!\cdots\!68}a^{5}-\frac{19\!\cdots\!31}{17\!\cdots\!67}a^{4}+\frac{21\!\cdots\!47}{46\!\cdots\!12}a^{3}-\frac{69428731927363}{761886908643129}a^{2}-\frac{782638853252287}{17\!\cdots\!67}a-\frac{15\!\cdots\!05}{58\!\cdots\!89}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$, $3$ |
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: |
\( -1 \)
(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: |
$\frac{34702017506817}{20\!\cdots\!44}a^{15}-\frac{109741712658311}{15\!\cdots\!58}a^{14}+\frac{114897751161961}{15\!\cdots\!58}a^{13}+\frac{983718309809803}{60\!\cdots\!32}a^{12}-\frac{238637202268358}{761886908643129}a^{11}-\frac{224737343391689}{761886908643129}a^{10}+\frac{30\!\cdots\!05}{60\!\cdots\!32}a^{9}+\frac{29412995576789}{11812200134002}a^{8}-\frac{44\!\cdots\!47}{10\!\cdots\!72}a^{7}-\frac{11\!\cdots\!33}{20\!\cdots\!44}a^{6}+\frac{30\!\cdots\!97}{15\!\cdots\!58}a^{5}-\frac{87\!\cdots\!22}{761886908643129}a^{4}-\frac{66\!\cdots\!85}{60\!\cdots\!32}a^{3}+\frac{90\!\cdots\!71}{10\!\cdots\!72}a^{2}+\frac{54\!\cdots\!55}{15\!\cdots\!58}a-\frac{12\!\cdots\!09}{761886908643129}$, $\frac{25\!\cdots\!47}{46\!\cdots\!12}a^{15}-\frac{13\!\cdots\!41}{70\!\cdots\!68}a^{14}+\frac{17\!\cdots\!47}{11\!\cdots\!78}a^{13}+\frac{75\!\cdots\!73}{14\!\cdots\!36}a^{12}-\frac{46\!\cdots\!11}{70\!\cdots\!68}a^{11}-\frac{64\!\cdots\!43}{58\!\cdots\!89}a^{10}+\frac{11\!\cdots\!11}{14\!\cdots\!36}a^{9}+\frac{42\!\cdots\!81}{543361206164092}a^{8}-\frac{21\!\cdots\!69}{23\!\cdots\!56}a^{7}-\frac{95\!\cdots\!71}{46\!\cdots\!12}a^{6}+\frac{35\!\cdots\!01}{70\!\cdots\!68}a^{5}-\frac{96\!\cdots\!49}{58\!\cdots\!89}a^{4}-\frac{40\!\cdots\!43}{14\!\cdots\!36}a^{3}+\frac{33\!\cdots\!89}{253962302881043}a^{2}+\frac{93\!\cdots\!12}{17\!\cdots\!67}a-\frac{34\!\cdots\!92}{17\!\cdots\!67}$, $\frac{111212041133281}{32\!\cdots\!52}a^{15}-\frac{112097703777089}{815041809246138}a^{14}+\frac{38291392914207}{271680603082046}a^{13}+\frac{331561307663939}{10\!\cdots\!84}a^{12}-\frac{228086639727356}{407520904623069}a^{11}-\frac{72392125311271}{135840301541023}a^{10}+\frac{26\!\cdots\!43}{32\!\cdots\!52}a^{9}+\frac{12\!\cdots\!13}{271680603082046}a^{8}-\frac{42\!\cdots\!73}{543361206164092}a^{7}-\frac{34\!\cdots\!61}{32\!\cdots\!52}a^{6}+\frac{30\!\cdots\!27}{815041809246138}a^{5}-\frac{30\!\cdots\!54}{135840301541023}a^{4}-\frac{51\!\cdots\!67}{32\!\cdots\!52}a^{3}+\frac{11\!\cdots\!93}{70873200804012}a^{2}+\frac{18\!\cdots\!31}{815041809246138}a-\frac{618843670717324}{407520904623069}$, $\frac{30\!\cdots\!67}{46\!\cdots\!12}a^{15}-\frac{16\!\cdots\!79}{70\!\cdots\!68}a^{14}+\frac{35\!\cdots\!56}{17\!\cdots\!67}a^{13}+\frac{82\!\cdots\!17}{14\!\cdots\!36}a^{12}-\frac{58\!\cdots\!93}{70\!\cdots\!68}a^{11}-\frac{38\!\cdots\!63}{35\!\cdots\!34}a^{10}+\frac{14\!\cdots\!39}{14\!\cdots\!36}a^{9}+\frac{48\!\cdots\!87}{543361206164092}a^{8}-\frac{27\!\cdots\!79}{23\!\cdots\!56}a^{7}-\frac{10\!\cdots\!91}{46\!\cdots\!12}a^{6}+\frac{43\!\cdots\!83}{70\!\cdots\!68}a^{5}-\frac{10\!\cdots\!01}{35\!\cdots\!34}a^{4}-\frac{35\!\cdots\!35}{14\!\cdots\!36}a^{3}+\frac{10\!\cdots\!75}{507924605762086}a^{2}-\frac{18\!\cdots\!15}{35\!\cdots\!34}a-\frac{15\!\cdots\!70}{17\!\cdots\!67}$, $\frac{20\!\cdots\!61}{70\!\cdots\!68}a^{15}-\frac{27\!\cdots\!93}{23\!\cdots\!56}a^{14}+\frac{46\!\cdots\!73}{35\!\cdots\!34}a^{13}+\frac{15\!\cdots\!21}{70\!\cdots\!68}a^{12}-\frac{10\!\cdots\!13}{23\!\cdots\!56}a^{11}-\frac{13\!\cdots\!33}{35\!\cdots\!34}a^{10}+\frac{13\!\cdots\!69}{23\!\cdots\!56}a^{9}+\frac{21\!\cdots\!11}{543361206164092}a^{8}-\frac{38\!\cdots\!93}{58\!\cdots\!89}a^{7}-\frac{53\!\cdots\!03}{70\!\cdots\!68}a^{6}+\frac{69\!\cdots\!93}{23\!\cdots\!56}a^{5}-\frac{79\!\cdots\!27}{35\!\cdots\!34}a^{4}-\frac{85\!\cdots\!29}{23\!\cdots\!56}a^{3}+\frac{23\!\cdots\!65}{30\!\cdots\!16}a^{2}+\frac{198424981506716}{17\!\cdots\!67}a-\frac{45\!\cdots\!99}{58\!\cdots\!89}$, $\frac{22\!\cdots\!11}{70\!\cdots\!68}a^{15}-\frac{79\!\cdots\!93}{70\!\cdots\!68}a^{14}+\frac{418340798932342}{58\!\cdots\!89}a^{13}+\frac{24\!\cdots\!07}{70\!\cdots\!68}a^{12}-\frac{27\!\cdots\!45}{70\!\cdots\!68}a^{11}-\frac{42\!\cdots\!77}{58\!\cdots\!89}a^{10}+\frac{12\!\cdots\!87}{23\!\cdots\!56}a^{9}+\frac{25\!\cdots\!09}{543361206164092}a^{8}-\frac{60\!\cdots\!43}{11\!\cdots\!78}a^{7}-\frac{92\!\cdots\!29}{70\!\cdots\!68}a^{6}+\frac{21\!\cdots\!21}{70\!\cdots\!68}a^{5}-\frac{32\!\cdots\!10}{58\!\cdots\!89}a^{4}-\frac{55\!\cdots\!55}{23\!\cdots\!56}a^{3}+\frac{32\!\cdots\!23}{30\!\cdots\!16}a^{2}+\frac{67\!\cdots\!83}{11\!\cdots\!78}a-\frac{15\!\cdots\!86}{58\!\cdots\!89}$, $\frac{4181144053}{126672387496}a^{15}-\frac{3055376565}{31668096874}a^{14}+\frac{160136673}{15834048437}a^{13}+\frac{46716459525}{126672387496}a^{12}-\frac{2590502822}{15834048437}a^{11}-\frac{28475205089}{31668096874}a^{10}-\frac{2530287709}{126672387496}a^{9}+\frac{158413685723}{31668096874}a^{8}-\frac{135912410917}{63336193748}a^{7}-\frac{1954992082449}{126672387496}a^{6}+\frac{673662356481}{31668096874}a^{5}+\frac{302604616681}{31668096874}a^{4}-\frac{2559608629355}{126672387496}a^{3}-\frac{307104673525}{63336193748}a^{2}+\frac{108714766887}{15834048437}a-\frac{15350415435}{15834048437}$
| 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: | \( 13880.0330176 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 13880.0330176 \cdot 1}{2\cdot\sqrt{1644508157944864993729}}\cr\approx \mathstrut & 0.415701349430 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 4*x^14 + 9*x^13 - 16*x^12 - 16*x^11 + 23*x^10 + 140*x^9 - 222*x^8 - 317*x^7 + 1076*x^6 - 632*x^5 - 455*x^4 + 462*x^3 + 36*x^2 - 88*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 - 4*x^15 + 4*x^14 + 9*x^13 - 16*x^12 - 16*x^11 + 23*x^10 + 140*x^9 - 222*x^8 - 317*x^7 + 1076*x^6 - 632*x^5 - 455*x^4 + 462*x^3 + 36*x^2 - 88*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 - 4*x^15 + 4*x^14 + 9*x^13 - 16*x^12 - 16*x^11 + 23*x^10 + 140*x^9 - 222*x^8 - 317*x^7 + 1076*x^6 - 632*x^5 - 455*x^4 + 462*x^3 + 36*x^2 - 88*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 - 4*x^15 + 4*x^14 + 9*x^13 - 16*x^12 - 16*x^11 + 23*x^10 + 140*x^9 - 222*x^8 - 317*x^7 + 1076*x^6 - 632*x^5 - 455*x^4 + 462*x^3 + 36*x^2 - 88*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
$C_2^3:C_4$ (as 16T33):
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 11 conjugacy class representatives for $C_2^3:C_4$ |
| Character table for $C_2^3:C_4$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{221}) \), 4.0.3757.1 x2, 4.0.2873.1 x2, \(\Q(\sqrt{13}, \sqrt{17})\), 8.4.40552535777.1 x2, 8.0.2385443281.1 x2, 8.0.2385443281.2 |
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 | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{4}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | R | R | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(13\)
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(17\)
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.4.3.1 | $x^{4} + 17$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 17.4.3.1 | $x^{4} + 17$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |