Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 3*x^13 + 23*x^12 + 5*x^11 - 26*x^10 - 67*x^9 - 59*x^8 - 7*x^7 + 123*x^6 + 210*x^5 + 236*x^4 + 174*x^3 + 104*x^2 + 30*x + 9)
gp: K = bnfinit(y^16 - 2*y^15 - 3*y^13 + 23*y^12 + 5*y^11 - 26*y^10 - 67*y^9 - 59*y^8 - 7*y^7 + 123*y^6 + 210*y^5 + 236*y^4 + 174*y^3 + 104*y^2 + 30*y + 9, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 3*x^13 + 23*x^12 + 5*x^11 - 26*x^10 - 67*x^9 - 59*x^8 - 7*x^7 + 123*x^6 + 210*x^5 + 236*x^4 + 174*x^3 + 104*x^2 + 30*x + 9);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 - 3*x^13 + 23*x^12 + 5*x^11 - 26*x^10 - 67*x^9 - 59*x^8 - 7*x^7 + 123*x^6 + 210*x^5 + 236*x^4 + 174*x^3 + 104*x^2 + 30*x + 9)
\( x^{16} - 2 x^{15} - 3 x^{13} + 23 x^{12} + 5 x^{11} - 26 x^{10} - 67 x^{9} - 59 x^{8} - 7 x^{7} + \cdots + 9 \)
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: |
\(30648714616416015625\)
\(\medspace = 5^{10}\cdot 11^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(16.52\) | 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: | $5^{3/4}11^{3/4}\approx 20.19630948441476$ | ||
| Ramified primes: |
\(5\), \(11\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{12}-\frac{1}{3}a^{11}-\frac{1}{3}a^{10}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a$, $\frac{1}{3}a^{14}+\frac{1}{3}a^{12}+\frac{1}{3}a^{10}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{433484755004541}a^{15}+\frac{28120912427258}{433484755004541}a^{14}+\frac{17581961091328}{433484755004541}a^{13}-\frac{177675020354008}{433484755004541}a^{12}+\frac{40377840293785}{433484755004541}a^{11}+\frac{61091129278151}{433484755004541}a^{10}-\frac{196590678870308}{433484755004541}a^{9}-\frac{13932357429407}{144494918334847}a^{8}+\frac{43298818633574}{144494918334847}a^{7}-\frac{150039657297949}{433484755004541}a^{6}-\frac{64890058479983}{144494918334847}a^{5}+\frac{70816809912140}{144494918334847}a^{4}+\frac{100542600353744}{433484755004541}a^{3}+\frac{77860023444947}{433484755004541}a^{2}-\frac{158147948854447}{433484755004541}a-\frac{45915071983780}{144494918334847}$
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 $1$
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: |
$\frac{9256892694757}{144494918334847}a^{15}-\frac{20305373103996}{144494918334847}a^{14}-\frac{14946445366478}{433484755004541}a^{13}-\frac{14951488350218}{433484755004541}a^{12}+\frac{629067792649589}{433484755004541}a^{11}+\frac{65535628482884}{433484755004541}a^{10}-\frac{458458633658596}{144494918334847}a^{9}-\frac{488544067378267}{144494918334847}a^{8}-\frac{319495013151809}{433484755004541}a^{7}+\frac{12\!\cdots\!70}{433484755004541}a^{6}+\frac{11\!\cdots\!49}{144494918334847}a^{5}+\frac{14\!\cdots\!12}{144494918334847}a^{4}+\frac{851828304050607}{144494918334847}a^{3}+\frac{333975886088876}{144494918334847}a^{2}+\frac{252257668221059}{433484755004541}a-\frac{70555326054822}{144494918334847}$, $\frac{27594814617529}{433484755004541}a^{15}-\frac{50698023780026}{433484755004541}a^{14}-\frac{18754125114587}{433484755004541}a^{13}-\frac{52893422935625}{433484755004541}a^{12}+\frac{586711082015554}{433484755004541}a^{11}+\frac{311599483508812}{433484755004541}a^{10}-\frac{982390656081065}{433484755004541}a^{9}-\frac{17\!\cdots\!24}{433484755004541}a^{8}-\frac{19\!\cdots\!14}{433484755004541}a^{7}+\frac{145424834126335}{433484755004541}a^{6}+\frac{12\!\cdots\!47}{144494918334847}a^{5}+\frac{20\!\cdots\!66}{144494918334847}a^{4}+\frac{65\!\cdots\!03}{433484755004541}a^{3}+\frac{15\!\cdots\!72}{144494918334847}a^{2}+\frac{20\!\cdots\!68}{433484755004541}a+\frac{251335639855252}{144494918334847}$, $\frac{2993888262337}{144494918334847}a^{15}+\frac{4445949888233}{144494918334847}a^{14}-\frac{75760584089420}{433484755004541}a^{13}-\frac{12921605763176}{433484755004541}a^{12}+\frac{155432943571967}{433484755004541}a^{11}+\frac{758533148051111}{433484755004541}a^{10}-\frac{108082018494013}{144494918334847}a^{9}-\frac{602705697807866}{144494918334847}a^{8}-\frac{19\!\cdots\!38}{433484755004541}a^{7}-\frac{821340516920542}{433484755004541}a^{6}+\frac{641181857704791}{144494918334847}a^{5}+\frac{19\!\cdots\!73}{144494918334847}a^{4}+\frac{22\!\cdots\!29}{144494918334847}a^{3}+\frac{16\!\cdots\!46}{144494918334847}a^{2}+\frac{26\!\cdots\!19}{433484755004541}a+\frac{331706011302412}{144494918334847}$, $\frac{30762117335047}{433484755004541}a^{15}-\frac{118822131777220}{433484755004541}a^{14}+\frac{160963779884221}{433484755004541}a^{13}-\frac{203878852500790}{433484755004541}a^{12}+\frac{915169276958398}{433484755004541}a^{11}-\frac{12\!\cdots\!92}{433484755004541}a^{10}-\frac{16997729256662}{433484755004541}a^{9}-\frac{230602068453584}{144494918334847}a^{8}+\frac{192800058710421}{144494918334847}a^{7}+\frac{944082275546693}{433484755004541}a^{6}+\frac{752484255083472}{144494918334847}a^{5}+\frac{121590196963560}{144494918334847}a^{4}+\frac{918895754296685}{433484755004541}a^{3}-\frac{10\!\cdots\!89}{433484755004541}a^{2}+\frac{322681149260441}{433484755004541}a-\frac{252412780571344}{144494918334847}$, $\frac{27266318946680}{433484755004541}a^{15}+\frac{3622814753553}{144494918334847}a^{14}-\frac{184789016122454}{433484755004541}a^{13}+\frac{15184466513993}{433484755004541}a^{12}+\frac{161081518402724}{144494918334847}a^{11}+\frac{17\!\cdots\!80}{433484755004541}a^{10}-\frac{16\!\cdots\!84}{433484755004541}a^{9}-\frac{41\!\cdots\!66}{433484755004541}a^{8}-\frac{39\!\cdots\!47}{433484755004541}a^{7}+\frac{96284295229474}{433484755004541}a^{6}+\frac{18\!\cdots\!36}{144494918334847}a^{5}+\frac{41\!\cdots\!72}{144494918334847}a^{4}+\frac{12\!\cdots\!48}{433484755004541}a^{3}+\frac{86\!\cdots\!04}{433484755004541}a^{2}+\frac{31\!\cdots\!76}{433484755004541}a+\frac{354430061749231}{144494918334847}$, $\frac{7791285886190}{144494918334847}a^{15}-\frac{62153443383149}{433484755004541}a^{14}+\frac{10813381829194}{144494918334847}a^{13}-\frac{76963209630728}{433484755004541}a^{12}+\frac{194858320671146}{144494918334847}a^{11}-\frac{206996097318737}{433484755004541}a^{10}-\frac{230304477160925}{144494918334847}a^{9}-\frac{12\!\cdots\!75}{433484755004541}a^{8}-\frac{566403122347616}{433484755004541}a^{7}+\frac{12\!\cdots\!52}{433484755004541}a^{6}+\frac{10\!\cdots\!63}{144494918334847}a^{5}+\frac{820187350651556}{144494918334847}a^{4}+\frac{629509547681875}{144494918334847}a^{3}+\frac{211922313767537}{433484755004541}a^{2}-\frac{107741590732895}{433484755004541}a-\frac{30255733091079}{144494918334847}$, $\frac{29098815302741}{433484755004541}a^{15}-\frac{16278646602950}{144494918334847}a^{14}-\frac{38786625496706}{433484755004541}a^{13}-\frac{53104923804982}{433484755004541}a^{12}+\frac{224025351191366}{144494918334847}a^{11}+\frac{380082552172237}{433484755004541}a^{10}-\frac{11\!\cdots\!86}{433484755004541}a^{9}-\frac{24\!\cdots\!87}{433484755004541}a^{8}-\frac{14\!\cdots\!16}{433484755004541}a^{7}+\frac{809184544820134}{433484755004541}a^{6}+\frac{14\!\cdots\!09}{144494918334847}a^{5}+\frac{21\!\cdots\!79}{144494918334847}a^{4}+\frac{57\!\cdots\!81}{433484755004541}a^{3}+\frac{30\!\cdots\!15}{433484755004541}a^{2}+\frac{14\!\cdots\!96}{433484755004541}a+\frac{113475181767460}{144494918334847}$
| 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: | \( 1255.64181615 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1255.64181615 \cdot 1}{2\cdot\sqrt{30648714616416015625}}\cr\approx \mathstrut & 0.275466371844 \end{aligned}\]
# 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^13 + 23*x^12 + 5*x^11 - 26*x^10 - 67*x^9 - 59*x^8 - 7*x^7 + 123*x^6 + 210*x^5 + 236*x^4 + 174*x^3 + 104*x^2 + 30*x + 9)
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^13 + 23*x^12 + 5*x^11 - 26*x^10 - 67*x^9 - 59*x^8 - 7*x^7 + 123*x^6 + 210*x^5 + 236*x^4 + 174*x^3 + 104*x^2 + 30*x + 9, 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^13 + 23*x^12 + 5*x^11 - 26*x^10 - 67*x^9 - 59*x^8 - 7*x^7 + 123*x^6 + 210*x^5 + 236*x^4 + 174*x^3 + 104*x^2 + 30*x + 9);
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^13 + 23*x^12 + 5*x^11 - 26*x^10 - 67*x^9 - 59*x^8 - 7*x^7 + 123*x^6 + 210*x^5 + 236*x^4 + 174*x^3 + 104*x^2 + 30*x + 9);
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
$C_4\wr C_2$ (as 16T42):
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 14 conjugacy class representatives for $C_4\wr C_2$ |
| Character table for $C_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-55}) \), 4.2.275.1 x2, 4.0.605.1 x2, \(\Q(\sqrt{5}, \sqrt{-11})\), 8.0.221445125.1, 8.0.5536128125.1, 8.0.9150625.1 |
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 8 siblings: | 8.0.221445125.1, 8.0.5536128125.1 |
| Degree 16 sibling: | 16.4.766217865410400390625.1 |
| Minimal sibling: | 8.0.221445125.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.8.6.2 | $x^{8} + 10 x^{4} - 25$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
|
\(11\)
| 11.8.6.1 | $x^{8} - 110 x^{4} - 16819$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |
| 11.8.6.1 | $x^{8} - 110 x^{4} - 16819$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |