Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 57*x^16 + 1314*x^14 + 15789*x^12 + 106527*x^10 + 406209*x^8 + 836429*x^6 + 820710*x^4 + 272034*x^2 + 16129)
gp: K = bnfinit(y^18 + 57*y^16 + 1314*y^14 + 15789*y^12 + 106527*y^10 + 406209*y^8 + 836429*y^6 + 820710*y^4 + 272034*y^2 + 16129, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 57*x^16 + 1314*x^14 + 15789*x^12 + 106527*x^10 + 406209*x^8 + 836429*x^6 + 820710*x^4 + 272034*x^2 + 16129);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 57*x^16 + 1314*x^14 + 15789*x^12 + 106527*x^10 + 406209*x^8 + 836429*x^6 + 820710*x^4 + 272034*x^2 + 16129)
\( x^{18} + 57 x^{16} + 1314 x^{14} + 15789 x^{12} + 106527 x^{10} + 406209 x^{8} + 836429 x^{6} + \cdots + 16129 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-16528519607100378938441688416256\)
\(\medspace = -\,2^{18}\cdot 3^{24}\cdot 7^{12}\cdot 127^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(54.24\) | 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: | $2^{7/4}3^{4/3}7^{2/3}127^{1/2}\approx 600.1567834330816$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(127\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{256}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{4}a^{12}+\frac{1}{4}a^{10}-\frac{1}{4}a^{8}+\frac{1}{4}a^{6}+\frac{1}{4}a^{4}-\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{4}a^{13}+\frac{1}{4}a^{11}-\frac{1}{4}a^{9}+\frac{1}{4}a^{7}+\frac{1}{4}a^{5}-\frac{1}{4}a^{3}+\frac{1}{4}a$, $\frac{1}{116}a^{14}-\frac{7}{58}a^{12}+\frac{10}{29}a^{10}+\frac{3}{29}a^{8}-\frac{19}{58}a^{6}-\frac{9}{29}a^{4}+\frac{10}{29}a^{2}+\frac{25}{116}$, $\frac{1}{116}a^{15}-\frac{7}{58}a^{13}+\frac{10}{29}a^{11}+\frac{3}{29}a^{9}-\frac{19}{58}a^{7}-\frac{9}{29}a^{5}+\frac{10}{29}a^{3}+\frac{25}{116}a$, $\frac{1}{13\!\cdots\!28}a^{16}-\frac{20678050345309}{13\!\cdots\!28}a^{14}-\frac{61\!\cdots\!99}{13\!\cdots\!28}a^{12}+\frac{38\!\cdots\!23}{13\!\cdots\!28}a^{10}-\frac{42\!\cdots\!85}{13\!\cdots\!28}a^{8}+\frac{27\!\cdots\!97}{13\!\cdots\!28}a^{6}-\frac{61\!\cdots\!09}{13\!\cdots\!28}a^{4}+\frac{10\!\cdots\!89}{66\!\cdots\!14}a^{2}-\frac{67825300453703}{261194721830491}$, $\frac{1}{13\!\cdots\!28}a^{17}-\frac{20678050345309}{13\!\cdots\!28}a^{15}-\frac{61\!\cdots\!99}{13\!\cdots\!28}a^{13}+\frac{38\!\cdots\!23}{13\!\cdots\!28}a^{11}-\frac{42\!\cdots\!85}{13\!\cdots\!28}a^{9}+\frac{27\!\cdots\!97}{13\!\cdots\!28}a^{7}-\frac{61\!\cdots\!09}{13\!\cdots\!28}a^{5}+\frac{10\!\cdots\!89}{66\!\cdots\!14}a^{3}-\frac{67825300453703}{261194721830491}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{252}$, which has order $2016$ (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: | $8$ | 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: |
$\frac{4108259544}{39443198183677}a^{16}+\frac{896408201337}{157772792734708}a^{14}+\frac{9716678174617}{78886396367354}a^{12}+\frac{53446610643675}{39443198183677}a^{10}+\frac{316558931815845}{39443198183677}a^{8}+\frac{19\!\cdots\!07}{78886396367354}a^{6}+\frac{15\!\cdots\!54}{39443198183677}a^{4}+\frac{899951899909653}{39443198183677}a^{2}+\frac{4111533794307}{1242305454604}$, $\frac{4140939693}{157772792734708}a^{16}+\frac{56723661546}{39443198183677}a^{14}+\frac{1233385961582}{39443198183677}a^{12}+\frac{13501693329285}{39443198183677}a^{10}+\frac{154812940189527}{78886396367354}a^{8}+\frac{213646151643892}{39443198183677}a^{6}+\frac{189091507006650}{39443198183677}a^{4}-\frac{457594495746471}{157772792734708}a^{2}-\frac{925086345533}{621152727302}$, $\frac{336547315}{14741353037428}a^{16}+\frac{9147292343}{7370676518714}a^{14}+\frac{98142286372}{3685338259357}a^{12}+\frac{1052585620383}{3685338259357}a^{10}+\frac{11737330406527}{7370676518714}a^{8}+\frac{15867561329364}{3685338259357}a^{6}+\frac{16483900790229}{3685338259357}a^{4}+\frac{21933312586919}{14741353037428}a^{2}+\frac{86114969241}{29018411491}$, $\frac{41075091}{508322518532}a^{16}+\frac{2267345613}{508322518532}a^{14}+\frac{24900486533}{254161259266}a^{12}+\frac{138692317530}{127080629633}a^{10}+\frac{1653138139123}{254161259266}a^{8}+\frac{5084281212389}{254161259266}a^{6}+\frac{3486496250463}{127080629633}a^{4}+\frac{5298582947883}{508322518532}a^{2}-\frac{3587445211}{4002539516}$, $\frac{5762108136385}{33\!\cdots\!57}a^{16}+\frac{627893190360355}{66\!\cdots\!14}a^{14}+\frac{13\!\cdots\!01}{66\!\cdots\!14}a^{12}+\frac{14\!\cdots\!05}{66\!\cdots\!14}a^{10}+\frac{84\!\cdots\!65}{66\!\cdots\!14}a^{8}+\frac{23\!\cdots\!81}{66\!\cdots\!14}a^{6}+\frac{27\!\cdots\!75}{66\!\cdots\!14}a^{4}+\frac{30\!\cdots\!71}{66\!\cdots\!14}a^{2}+\frac{237517782675063}{261194721830491}$, $\frac{1953763010302}{33\!\cdots\!57}a^{16}+\frac{212000186289291}{66\!\cdots\!14}a^{14}+\frac{90\!\cdots\!73}{13\!\cdots\!28}a^{12}+\frac{97\!\cdots\!61}{13\!\cdots\!28}a^{10}+\frac{55\!\cdots\!71}{13\!\cdots\!28}a^{8}+\frac{16\!\cdots\!77}{13\!\cdots\!28}a^{6}+\frac{20\!\cdots\!33}{13\!\cdots\!28}a^{4}+\frac{91\!\cdots\!83}{13\!\cdots\!28}a^{2}+\frac{17\!\cdots\!57}{10\!\cdots\!64}$, $\frac{354107223098}{33\!\cdots\!57}a^{16}+\frac{14201007685612}{33\!\cdots\!57}a^{14}+\frac{347482480675451}{66\!\cdots\!14}a^{12}+\frac{375945398443305}{66\!\cdots\!14}a^{10}-\frac{19\!\cdots\!49}{66\!\cdots\!14}a^{8}-\frac{11\!\cdots\!73}{66\!\cdots\!14}a^{6}-\frac{19\!\cdots\!55}{66\!\cdots\!14}a^{4}-\frac{43\!\cdots\!67}{66\!\cdots\!14}a^{2}+\frac{246776400959927}{522389443660982}$, $\frac{5082994442536}{33\!\cdots\!57}a^{16}+\frac{11\!\cdots\!47}{13\!\cdots\!28}a^{14}+\frac{12\!\cdots\!65}{66\!\cdots\!14}a^{12}+\frac{65\!\cdots\!43}{33\!\cdots\!57}a^{10}+\frac{38\!\cdots\!12}{33\!\cdots\!57}a^{8}+\frac{23\!\cdots\!59}{66\!\cdots\!14}a^{6}+\frac{15\!\cdots\!93}{33\!\cdots\!57}a^{4}+\frac{71\!\cdots\!25}{33\!\cdots\!57}a^{2}+\frac{29\!\cdots\!17}{10\!\cdots\!64}$
| 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: | \( 54408.4888887 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 54408.4888887 \cdot 2016}{2\cdot\sqrt{16528519607100378938441688416256}}\cr\approx \mathstrut & 0.205887050910 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 + 57*x^16 + 1314*x^14 + 15789*x^12 + 106527*x^10 + 406209*x^8 + 836429*x^6 + 820710*x^4 + 272034*x^2 + 16129)
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^18 + 57*x^16 + 1314*x^14 + 15789*x^12 + 106527*x^10 + 406209*x^8 + 836429*x^6 + 820710*x^4 + 272034*x^2 + 16129, 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^18 + 57*x^16 + 1314*x^14 + 15789*x^12 + 106527*x^10 + 406209*x^8 + 836429*x^6 + 820710*x^4 + 272034*x^2 + 16129);
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^18 + 57*x^16 + 1314*x^14 + 15789*x^12 + 106527*x^10 + 406209*x^8 + 836429*x^6 + 820710*x^4 + 272034*x^2 + 16129);
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
$C_2^3:A_4^2$ (as 18T263):
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 1152 |
| The 48 conjugacy class representatives for $C_2^3:A_4^2$ |
| Character table for $C_2^3:A_4^2$ |
Intermediate fields
| 3.3.3969.1, \(\Q(\zeta_{9})^+\), 3.3.3969.2, \(\Q(\zeta_{7})^+\), 9.9.62523502209.1 |
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
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
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 | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.6.4 | $x^{6} - 4 x^{5} + 14 x^{4} - 24 x^{3} + 100 x^{2} + 48 x + 88$ | $2$ | $3$ | $6$ | $A_4\times C_2$ | $[2, 2, 2]^{3}$ |
| 2.6.6.4 | $x^{6} - 4 x^{5} + 14 x^{4} - 24 x^{3} + 100 x^{2} + 48 x + 88$ | $2$ | $3$ | $6$ | $A_4\times C_2$ | $[2, 2, 2]^{3}$ | |
| 2.6.6.4 | $x^{6} - 4 x^{5} + 14 x^{4} - 24 x^{3} + 100 x^{2} + 48 x + 88$ | $2$ | $3$ | $6$ | $A_4\times C_2$ | $[2, 2, 2]^{3}$ | |
|
\(3\)
| Deg $18$ | $3$ | $6$ | $24$ | |||
|
\(7\)
| 7.18.12.1 | $x^{18} + 3 x^{16} + 57 x^{15} + 15 x^{14} + 90 x^{13} + 424 x^{12} - 921 x^{11} - 3090 x^{10} - 6496 x^{9} - 10560 x^{8} + 6912 x^{7} + 28033 x^{6} + 33237 x^{5} + 188463 x^{4} - 139476 x^{3} + 351552 x^{2} - 514905 x + 582014$ | $3$ | $6$ | $12$ | $C_6 \times C_3$ | $[\ ]_{3}^{6}$ |
|
\(127\)
| $\Q_{127}$ | $x + 124$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{127}$ | $x + 124$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{127}$ | $x + 124$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{127}$ | $x + 124$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{127}$ | $x + 124$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{127}$ | $x + 124$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{127}$ | $x + 124$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{127}$ | $x + 124$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 127.2.1.2 | $x^{2} + 127$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 127.2.1.2 | $x^{2} + 127$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 127.2.0.1 | $x^{2} + 126 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 127.2.0.1 | $x^{2} + 126 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 127.2.0.1 | $x^{2} + 126 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |