Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 3*x^14 + 67*x^13 + 19*x^12 - 142*x^11 + 792*x^10 + 1862*x^9 + 7725*x^8 + 6707*x^7 - 17082*x^6 + 2780*x^5 + 54986*x^4 + 52783*x^3 + 32352*x^2 - 2819*x + 11519)
gp: K = bnfinit(y^16 - 2*y^15 + 3*y^14 + 67*y^13 + 19*y^12 - 142*y^11 + 792*y^10 + 1862*y^9 + 7725*y^8 + 6707*y^7 - 17082*y^6 + 2780*y^5 + 54986*y^4 + 52783*y^3 + 32352*y^2 - 2819*y + 11519, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 3*x^14 + 67*x^13 + 19*x^12 - 142*x^11 + 792*x^10 + 1862*x^9 + 7725*x^8 + 6707*x^7 - 17082*x^6 + 2780*x^5 + 54986*x^4 + 52783*x^3 + 32352*x^2 - 2819*x + 11519);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 + 3*x^14 + 67*x^13 + 19*x^12 - 142*x^11 + 792*x^10 + 1862*x^9 + 7725*x^8 + 6707*x^7 - 17082*x^6 + 2780*x^5 + 54986*x^4 + 52783*x^3 + 32352*x^2 - 2819*x + 11519)
\( x^{16} - 2 x^{15} + 3 x^{14} + 67 x^{13} + 19 x^{12} - 142 x^{11} + 792 x^{10} + 1862 x^{9} + \cdots + 11519 \)
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: |
\(11059281013440062648863435321\)
\(\medspace = 7^{12}\cdot 19^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(56.59\) | 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: | $7^{3/4}19^{7/8}\approx 56.58911875386272$ | ||
| Ramified primes: |
\(7\), \(19\)
| 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 over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{63}a^{12}+\frac{1}{9}a^{11}-\frac{1}{7}a^{10}+\frac{4}{63}a^{9}+\frac{5}{63}a^{8}-\frac{3}{7}a^{7}+\frac{10}{21}a^{6}+\frac{1}{21}a^{5}+\frac{5}{63}a^{4}-\frac{1}{9}a^{3}-\frac{10}{21}a^{2}-\frac{16}{63}a-\frac{20}{63}$, $\frac{1}{2457}a^{13}+\frac{1}{819}a^{12}+\frac{5}{2457}a^{11}+\frac{124}{2457}a^{10}-\frac{32}{2457}a^{9}+\frac{79}{2457}a^{8}-\frac{353}{819}a^{7}-\frac{20}{273}a^{6}+\frac{116}{351}a^{5}-\frac{73}{273}a^{4}+\frac{523}{2457}a^{3}+\frac{335}{2457}a^{2}-\frac{880}{2457}a+\frac{899}{2457}$, $\frac{1}{100737}a^{14}-\frac{5}{100737}a^{13}-\frac{643}{100737}a^{12}+\frac{49}{1599}a^{11}+\frac{16}{351}a^{10}+\frac{5210}{100737}a^{9}+\frac{14026}{100737}a^{8}+\frac{1828}{33579}a^{7}+\frac{40043}{100737}a^{6}-\frac{40966}{100737}a^{5}-\frac{21911}{100737}a^{4}-\frac{13750}{33579}a^{3}-\frac{9410}{100737}a^{2}+\frac{42493}{100737}a-\frac{8635}{100737}$, $\frac{1}{63\!\cdots\!93}a^{15}+\frac{59\!\cdots\!83}{21\!\cdots\!31}a^{14}-\frac{80\!\cdots\!55}{49\!\cdots\!17}a^{13}+\frac{29\!\cdots\!58}{63\!\cdots\!93}a^{12}-\frac{11\!\cdots\!36}{21\!\cdots\!31}a^{11}+\frac{10\!\cdots\!23}{63\!\cdots\!93}a^{10}+\frac{60\!\cdots\!52}{63\!\cdots\!93}a^{9}+\frac{64\!\cdots\!29}{63\!\cdots\!93}a^{8}+\frac{11\!\cdots\!64}{63\!\cdots\!93}a^{7}+\frac{88\!\cdots\!47}{21\!\cdots\!31}a^{6}+\frac{44\!\cdots\!69}{21\!\cdots\!31}a^{5}+\frac{18\!\cdots\!97}{63\!\cdots\!93}a^{4}-\frac{44\!\cdots\!61}{10\!\cdots\!11}a^{3}-\frac{25\!\cdots\!23}{63\!\cdots\!93}a^{2}-\frac{31\!\cdots\!39}{63\!\cdots\!93}a-\frac{16\!\cdots\!07}{63\!\cdots\!93}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
Trivial group, which has order $1$ (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: |
$\frac{23\!\cdots\!32}{21\!\cdots\!93}a^{15}-\frac{25\!\cdots\!23}{49\!\cdots\!17}a^{14}+\frac{33\!\cdots\!01}{70\!\cdots\!31}a^{13}+\frac{10\!\cdots\!10}{14\!\cdots\!51}a^{12}-\frac{90\!\cdots\!81}{49\!\cdots\!17}a^{11}-\frac{11\!\cdots\!86}{21\!\cdots\!93}a^{10}+\frac{11\!\cdots\!33}{14\!\cdots\!51}a^{9}-\frac{79\!\cdots\!50}{14\!\cdots\!51}a^{8}+\frac{37\!\cdots\!20}{14\!\cdots\!51}a^{7}-\frac{13\!\cdots\!37}{49\!\cdots\!17}a^{6}-\frac{11\!\cdots\!31}{12\!\cdots\!37}a^{5}-\frac{84\!\cdots\!83}{16\!\cdots\!61}a^{4}+\frac{27\!\cdots\!81}{54\!\cdots\!13}a^{3}-\frac{57\!\cdots\!63}{14\!\cdots\!51}a^{2}+\frac{15\!\cdots\!13}{14\!\cdots\!51}a-\frac{16\!\cdots\!04}{14\!\cdots\!51}$, $\frac{95\!\cdots\!00}{21\!\cdots\!93}a^{15}-\frac{54\!\cdots\!99}{49\!\cdots\!17}a^{14}+\frac{92\!\cdots\!83}{49\!\cdots\!17}a^{13}+\frac{43\!\cdots\!54}{14\!\cdots\!51}a^{12}-\frac{27\!\cdots\!44}{49\!\cdots\!17}a^{11}-\frac{94\!\cdots\!55}{14\!\cdots\!51}a^{10}+\frac{59\!\cdots\!25}{14\!\cdots\!51}a^{9}+\frac{96\!\cdots\!59}{14\!\cdots\!51}a^{8}+\frac{44\!\cdots\!01}{14\!\cdots\!51}a^{7}+\frac{13\!\cdots\!85}{70\!\cdots\!31}a^{6}-\frac{52\!\cdots\!25}{70\!\cdots\!31}a^{5}+\frac{61\!\cdots\!52}{14\!\cdots\!51}a^{4}+\frac{11\!\cdots\!92}{54\!\cdots\!13}a^{3}+\frac{27\!\cdots\!70}{21\!\cdots\!93}a^{2}+\frac{19\!\cdots\!43}{14\!\cdots\!51}a-\frac{45\!\cdots\!10}{14\!\cdots\!51}$, $\frac{26\!\cdots\!69}{14\!\cdots\!51}a^{15}-\frac{12\!\cdots\!87}{49\!\cdots\!17}a^{14}+\frac{34\!\cdots\!28}{49\!\cdots\!17}a^{13}+\frac{19\!\cdots\!81}{14\!\cdots\!51}a^{12}+\frac{43\!\cdots\!26}{49\!\cdots\!17}a^{11}-\frac{46\!\cdots\!95}{11\!\cdots\!27}a^{10}+\frac{20\!\cdots\!36}{14\!\cdots\!51}a^{9}+\frac{66\!\cdots\!43}{14\!\cdots\!51}a^{8}+\frac{19\!\cdots\!15}{14\!\cdots\!51}a^{7}+\frac{79\!\cdots\!83}{49\!\cdots\!17}a^{6}-\frac{19\!\cdots\!03}{49\!\cdots\!17}a^{5}-\frac{23\!\cdots\!60}{14\!\cdots\!51}a^{4}+\frac{28\!\cdots\!97}{18\!\cdots\!29}a^{3}+\frac{20\!\cdots\!64}{21\!\cdots\!93}a^{2}-\frac{48\!\cdots\!70}{14\!\cdots\!51}a+\frac{55\!\cdots\!11}{14\!\cdots\!51}$, $\frac{16\!\cdots\!47}{63\!\cdots\!93}a^{15}-\frac{13\!\cdots\!78}{21\!\cdots\!31}a^{14}+\frac{54\!\cdots\!24}{49\!\cdots\!17}a^{13}+\frac{10\!\cdots\!76}{63\!\cdots\!93}a^{12}-\frac{99\!\cdots\!59}{21\!\cdots\!31}a^{11}-\frac{17\!\cdots\!23}{48\!\cdots\!61}a^{10}+\frac{22\!\cdots\!17}{90\!\cdots\!99}a^{9}+\frac{17\!\cdots\!76}{63\!\cdots\!93}a^{8}+\frac{17\!\cdots\!36}{90\!\cdots\!99}a^{7}+\frac{36\!\cdots\!90}{30\!\cdots\!33}a^{6}-\frac{11\!\cdots\!02}{21\!\cdots\!31}a^{5}+\frac{24\!\cdots\!40}{15\!\cdots\!73}a^{4}+\frac{47\!\cdots\!49}{33\!\cdots\!37}a^{3}+\frac{68\!\cdots\!94}{63\!\cdots\!93}a^{2}+\frac{26\!\cdots\!74}{90\!\cdots\!99}a-\frac{42\!\cdots\!83}{63\!\cdots\!93}$, $\frac{14\!\cdots\!61}{63\!\cdots\!93}a^{15}-\frac{18\!\cdots\!00}{21\!\cdots\!31}a^{14}+\frac{69\!\cdots\!36}{49\!\cdots\!17}a^{13}+\frac{96\!\cdots\!90}{63\!\cdots\!93}a^{12}-\frac{54\!\cdots\!14}{21\!\cdots\!31}a^{11}-\frac{32\!\cdots\!64}{63\!\cdots\!93}a^{10}+\frac{21\!\cdots\!31}{63\!\cdots\!93}a^{9}+\frac{92\!\cdots\!40}{63\!\cdots\!93}a^{8}+\frac{21\!\cdots\!26}{48\!\cdots\!61}a^{7}-\frac{70\!\cdots\!80}{16\!\cdots\!87}a^{6}-\frac{20\!\cdots\!03}{30\!\cdots\!33}a^{5}+\frac{23\!\cdots\!61}{63\!\cdots\!93}a^{4}+\frac{17\!\cdots\!03}{10\!\cdots\!11}a^{3}+\frac{30\!\cdots\!94}{90\!\cdots\!99}a^{2}-\frac{62\!\cdots\!96}{48\!\cdots\!61}a+\frac{31\!\cdots\!75}{90\!\cdots\!99}$, $\frac{18\!\cdots\!88}{63\!\cdots\!93}a^{15}+\frac{17\!\cdots\!03}{21\!\cdots\!31}a^{14}+\frac{65\!\cdots\!96}{49\!\cdots\!17}a^{13}+\frac{14\!\cdots\!68}{63\!\cdots\!93}a^{12}+\frac{21\!\cdots\!82}{21\!\cdots\!31}a^{11}+\frac{13\!\cdots\!55}{63\!\cdots\!93}a^{10}+\frac{27\!\cdots\!37}{63\!\cdots\!93}a^{9}+\frac{14\!\cdots\!52}{90\!\cdots\!99}a^{8}+\frac{10\!\cdots\!90}{15\!\cdots\!73}a^{7}+\frac{44\!\cdots\!63}{21\!\cdots\!31}a^{6}+\frac{84\!\cdots\!79}{21\!\cdots\!31}a^{5}+\frac{27\!\cdots\!20}{63\!\cdots\!93}a^{4}+\frac{18\!\cdots\!46}{70\!\cdots\!77}a^{3}+\frac{67\!\cdots\!26}{63\!\cdots\!93}a^{2}+\frac{22\!\cdots\!57}{63\!\cdots\!93}a+\frac{19\!\cdots\!91}{63\!\cdots\!93}$, $\frac{15\!\cdots\!93}{63\!\cdots\!93}a^{15}-\frac{26\!\cdots\!61}{21\!\cdots\!31}a^{14}+\frac{28\!\cdots\!08}{70\!\cdots\!31}a^{13}+\frac{54\!\cdots\!63}{63\!\cdots\!93}a^{12}-\frac{78\!\cdots\!53}{21\!\cdots\!31}a^{11}+\frac{57\!\cdots\!35}{63\!\cdots\!93}a^{10}-\frac{17\!\cdots\!75}{63\!\cdots\!93}a^{9}+\frac{25\!\cdots\!92}{69\!\cdots\!23}a^{8}+\frac{61\!\cdots\!88}{63\!\cdots\!93}a^{7}-\frac{40\!\cdots\!80}{21\!\cdots\!31}a^{6}+\frac{23\!\cdots\!64}{21\!\cdots\!31}a^{5}+\frac{16\!\cdots\!21}{63\!\cdots\!93}a^{4}+\frac{40\!\cdots\!46}{33\!\cdots\!37}a^{3}+\frac{42\!\cdots\!00}{63\!\cdots\!93}a^{2}-\frac{19\!\cdots\!01}{90\!\cdots\!99}a+\frac{12\!\cdots\!58}{63\!\cdots\!93}$
| 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: | \( 25363590.094 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 25363590.094 \cdot 1}{2\cdot\sqrt{11059281013440062648863435321}}\cr\approx \mathstrut & 0.29292485951 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 3*x^14 + 67*x^13 + 19*x^12 - 142*x^11 + 792*x^10 + 1862*x^9 + 7725*x^8 + 6707*x^7 - 17082*x^6 + 2780*x^5 + 54986*x^4 + 52783*x^3 + 32352*x^2 - 2819*x + 11519)
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 - 2*x^15 + 3*x^14 + 67*x^13 + 19*x^12 - 142*x^11 + 792*x^10 + 1862*x^9 + 7725*x^8 + 6707*x^7 - 17082*x^6 + 2780*x^5 + 54986*x^4 + 52783*x^3 + 32352*x^2 - 2819*x + 11519, 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 - 2*x^15 + 3*x^14 + 67*x^13 + 19*x^12 - 142*x^11 + 792*x^10 + 1862*x^9 + 7725*x^8 + 6707*x^7 - 17082*x^6 + 2780*x^5 + 54986*x^4 + 52783*x^3 + 32352*x^2 - 2819*x + 11519);
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 - 2*x^15 + 3*x^14 + 67*x^13 + 19*x^12 - 142*x^11 + 792*x^10 + 1862*x^9 + 7725*x^8 + 6707*x^7 - 17082*x^6 + 2780*x^5 + 54986*x^4 + 52783*x^3 + 32352*x^2 - 2819*x + 11519);
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
$\SD_{16}$ (as 16T12):
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 16 |
| The 7 conjugacy class representatives for $QD_{16}$ |
| Character table for $QD_{16}$ |
Intermediate fields
| \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{133}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{-7}, \sqrt{-19})\), 4.2.336091.1 x2, 4.0.48013.1 x2, 8.0.112957160281.1, 8.2.105163116221611.1 x4 |
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 | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.2.0.1}{2} }^{8}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.8.6.1 | $x^{8} + 14 x^{4} - 245$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |
| 7.8.6.1 | $x^{8} + 14 x^{4} - 245$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ | |
|
\(19\)
| 19.16.14.1 | $x^{16} - 5396 x^{8} - 21660$ | $8$ | $2$ | $14$ | $QD_{16}$ | $[\ ]_{8}^{2}$ |