Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 9*x^16 - 9*x^15 + 36*x^14 - 63*x^13 + 109*x^12 - 189*x^11 + 252*x^10 - 335*x^9 + 378*x^8 - 378*x^7 + 334*x^6 - 243*x^5 + 147*x^4 - 67*x^3 + 21*x^2 - 3*x + 1)
gp: K = bnfinit(y^18 + 9*y^16 - 9*y^15 + 36*y^14 - 63*y^13 + 109*y^12 - 189*y^11 + 252*y^10 - 335*y^9 + 378*y^8 - 378*y^7 + 334*y^6 - 243*y^5 + 147*y^4 - 67*y^3 + 21*y^2 - 3*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 9*x^16 - 9*x^15 + 36*x^14 - 63*x^13 + 109*x^12 - 189*x^11 + 252*x^10 - 335*x^9 + 378*x^8 - 378*x^7 + 334*x^6 - 243*x^5 + 147*x^4 - 67*x^3 + 21*x^2 - 3*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 9*x^16 - 9*x^15 + 36*x^14 - 63*x^13 + 109*x^12 - 189*x^11 + 252*x^10 - 335*x^9 + 378*x^8 - 378*x^7 + 334*x^6 - 243*x^5 + 147*x^4 - 67*x^3 + 21*x^2 - 3*x + 1)
\( x^{18} + 9 x^{16} - 9 x^{15} + 36 x^{14} - 63 x^{13} + 109 x^{12} - 189 x^{11} + 252 x^{10} - 335 x^{9} + \cdots + 1 \)
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: |
\(-1548507685660102467\)
\(\medspace = -\,3^{21}\cdot 23^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(10.25\) | 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: | $3^{25/18}23^{1/2}\approx 22.05628309434856$ | ||
| Ramified primes: |
\(3\), \(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{1153182335}a^{17}-\frac{394336921}{1153182335}a^{16}-\frac{85928896}{230636467}a^{15}-\frac{97238009}{1153182335}a^{14}-\frac{49031889}{230636467}a^{13}-\frac{138175543}{1153182335}a^{12}+\frac{462712092}{1153182335}a^{11}-\frac{279666761}{1153182335}a^{10}-\frac{280014797}{1153182335}a^{9}-\frac{524613583}{1153182335}a^{8}+\frac{63869246}{1153182335}a^{7}-\frac{17051817}{67834255}a^{6}-\frac{77106787}{1153182335}a^{5}-\frac{214113886}{1153182335}a^{4}+\frac{560320823}{1153182335}a^{3}-\frac{96385234}{230636467}a^{2}-\frac{142185499}{1153182335}a+\frac{564604746}{1153182335}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
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: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{3077572}{11195945} a^{17} - \frac{313528}{11195945} a^{16} - \frac{5796705}{2239189} a^{15} + \frac{22523788}{11195945} a^{14} - \frac{24327282}{2239189} a^{13} + \frac{172258676}{11195945} a^{12} - \frac{359270994}{11195945} a^{11} + \frac{555197617}{11195945} a^{10} - \frac{776133451}{11195945} a^{9} + \frac{1033752351}{11195945} a^{8} - \frac{1120967627}{11195945} a^{7} + \frac{68522744}{658585} a^{6} - \frac{989519121}{11195945} a^{5} + \frac{715640877}{11195945} a^{4} - \frac{427450321}{11195945} a^{3} + \frac{36791096}{2239189} a^{2} - \frac{55256652}{11195945} a + \frac{15041843}{11195945} \)
(order $6$)
| 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: |
$\frac{7787269}{67834255}a^{17}-\frac{775114}{67834255}a^{16}+\frac{14561708}{13566851}a^{15}-\frac{75084051}{67834255}a^{14}+\frac{63477544}{13566851}a^{13}-\frac{513110897}{67834255}a^{12}+\frac{1040788863}{67834255}a^{11}-\frac{1598219739}{67834255}a^{10}+\frac{2471706147}{67834255}a^{9}-\frac{3126190612}{67834255}a^{8}+\frac{3731901739}{67834255}a^{7}-\frac{3993706766}{67834255}a^{6}+\frac{3491474512}{67834255}a^{5}-\frac{2883271559}{67834255}a^{4}+\frac{1815929392}{67834255}a^{3}-\frac{178925970}{13566851}a^{2}+\frac{312293934}{67834255}a-\frac{27417146}{67834255}$, $\frac{113222024}{1153182335}a^{17}+\frac{55131366}{1153182335}a^{16}+\frac{193757382}{230636467}a^{15}-\frac{543260141}{1153182335}a^{14}+\frac{622325566}{230636467}a^{13}-\frac{4879861732}{1153182335}a^{12}+\frac{7328683703}{1153182335}a^{11}-\frac{12523178434}{1153182335}a^{10}+\frac{15206422027}{1153182335}a^{9}-\frac{15405478787}{1153182335}a^{8}+\frac{17985332554}{1153182335}a^{7}-\frac{666095633}{67834255}a^{6}+\frac{5748851982}{1153182335}a^{5}-\frac{2073417299}{1153182335}a^{4}-\frac{4448509843}{1153182335}a^{3}+\frac{793439782}{230636467}a^{2}-\frac{1950402161}{1153182335}a+\frac{340926974}{1153182335}$, $\frac{178802317}{1153182335}a^{17}+\frac{50157738}{1153182335}a^{16}+\frac{295167334}{230636467}a^{15}-\frac{1158575403}{1153182335}a^{14}+\frac{987272678}{230636467}a^{13}-\frac{8311404001}{1153182335}a^{12}+\frac{12673519399}{1153182335}a^{11}-\frac{21722715152}{1153182335}a^{10}+\frac{25092904776}{1153182335}a^{9}-\frac{31254712571}{1153182335}a^{8}+\frac{29876349352}{1153182335}a^{7}-\frac{1475263934}{67834255}a^{6}+\frac{17684984381}{1153182335}a^{5}-\frac{6013754842}{1153182335}a^{4}+\frac{1025853856}{1153182335}a^{3}+\frac{220696501}{230636467}a^{2}-\frac{2005756463}{1153182335}a+\frac{388091067}{1153182335}$, $\frac{26858599}{1153182335}a^{17}+\frac{192380726}{1153182335}a^{16}+\frac{8888358}{230636467}a^{15}+\frac{1074178204}{1153182335}a^{14}-\frac{537387521}{230636467}a^{13}+\frac{3331740458}{1153182335}a^{12}-\frac{14050528177}{1153182335}a^{11}+\frac{14856338311}{1153182335}a^{10}-\frac{32487491478}{1153182335}a^{9}+\frac{41195931338}{1153182335}a^{8}-\frac{49412987666}{1153182335}a^{7}+\frac{3401221482}{67834255}a^{6}-\frac{49307725013}{1153182335}a^{5}+\frac{41730658171}{1153182335}a^{4}-\frac{23852906973}{1153182335}a^{3}+\frac{2061584994}{230636467}a^{2}-\frac{2770205536}{1153182335}a+\frac{613268979}{1153182335}$, $\frac{335539153}{1153182335}a^{17}+\frac{415907127}{1153182335}a^{16}+\frac{617166644}{230636467}a^{15}+\frac{449056538}{1153182335}a^{14}+\frac{1751882503}{230636467}a^{13}-\frac{9039130324}{1153182335}a^{12}+\frac{13975578746}{1153182335}a^{11}-\frac{28814422013}{1153182335}a^{10}+\frac{24996012669}{1153182335}a^{9}-\frac{34483388644}{1153182335}a^{8}+\frac{34033375843}{1153182335}a^{7}-\frac{1312022291}{67834255}a^{6}+\frac{16964605959}{1153182335}a^{5}-\frac{6132877753}{1153182335}a^{4}-\frac{2344263796}{1153182335}a^{3}+\frac{486836096}{230636467}a^{2}-\frac{3113200302}{1153182335}a+\frac{1680560288}{1153182335}$, $\frac{7717972}{67834255}a^{17}+\frac{13555223}{67834255}a^{16}+\frac{16262261}{13566851}a^{15}+\frac{47173707}{67834255}a^{14}+\frac{49157180}{13566851}a^{13}-\frac{167600876}{67834255}a^{12}+\frac{308768684}{67834255}a^{11}-\frac{895490907}{67834255}a^{10}+\frac{497754946}{67834255}a^{9}-\frac{1423818216}{67834255}a^{8}+\frac{1188695972}{67834255}a^{7}-\frac{1297973193}{67834255}a^{6}+\frac{1421199251}{67834255}a^{5}-\frac{920550427}{67834255}a^{4}+\frac{845243156}{67834255}a^{3}-\frac{67781282}{13566851}a^{2}+\frac{211776622}{67834255}a-\frac{901808}{67834255}$, $\frac{94166316}{1153182335}a^{17}+\frac{237634764}{1153182335}a^{16}+\frac{258123870}{230636467}a^{15}+\frac{1727320226}{1153182335}a^{14}+\frac{1055059070}{230636467}a^{13}+\frac{2443434597}{1153182335}a^{12}+\frac{7905724342}{1153182335}a^{11}-\frac{5121191536}{1153182335}a^{10}+\frac{4278272468}{1153182335}a^{9}-\frac{13311177608}{1153182335}a^{8}+\frac{2541094371}{1153182335}a^{7}-\frac{374200542}{67834255}a^{6}+\frac{588237258}{1153182335}a^{5}+\frac{3665657139}{1153182335}a^{4}-\frac{4964064202}{1153182335}a^{3}+\frac{1171730599}{230636467}a^{2}-\frac{3469817434}{1153182335}a+\frac{417084781}{1153182335}$, $\frac{305725712}{1153182335}a^{17}+\frac{412187843}{1153182335}a^{16}+\frac{616491220}{230636467}a^{15}+\frac{1183190987}{1153182335}a^{14}+\frac{2097113568}{230636467}a^{13}-\frac{5066018986}{1153182335}a^{12}+\frac{19483233154}{1153182335}a^{11}-\frac{24589697202}{1153182335}a^{10}+\frac{29036839026}{1153182335}a^{9}-\frac{40044302106}{1153182335}a^{8}+\frac{34383968397}{1153182335}a^{7}-\frac{1960080519}{67834255}a^{6}+\frac{21977459606}{1153182335}a^{5}-\frac{13295745582}{1153182335}a^{4}+\frac{3183462736}{1153182335}a^{3}-\frac{339693754}{230636467}a^{2}-\frac{1272504488}{1153182335}a+\frac{206732022}{1153182335}$
| 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: | \( 78.7017437829 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 78.7017437829 \cdot 1}{6\cdot\sqrt{1548507685660102467}}\cr\approx \mathstrut & 0.160877473511 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 + 9*x^16 - 9*x^15 + 36*x^14 - 63*x^13 + 109*x^12 - 189*x^11 + 252*x^10 - 335*x^9 + 378*x^8 - 378*x^7 + 334*x^6 - 243*x^5 + 147*x^4 - 67*x^3 + 21*x^2 - 3*x + 1)
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 + 9*x^16 - 9*x^15 + 36*x^14 - 63*x^13 + 109*x^12 - 189*x^11 + 252*x^10 - 335*x^9 + 378*x^8 - 378*x^7 + 334*x^6 - 243*x^5 + 147*x^4 - 67*x^3 + 21*x^2 - 3*x + 1, 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 + 9*x^16 - 9*x^15 + 36*x^14 - 63*x^13 + 109*x^12 - 189*x^11 + 252*x^10 - 335*x^9 + 378*x^8 - 378*x^7 + 334*x^6 - 243*x^5 + 147*x^4 - 67*x^3 + 21*x^2 - 3*x + 1);
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 + 9*x^16 - 9*x^15 + 36*x^14 - 63*x^13 + 109*x^12 - 189*x^11 + 252*x^10 - 335*x^9 + 378*x^8 - 378*x^7 + 334*x^6 - 243*x^5 + 147*x^4 - 67*x^3 + 21*x^2 - 3*x + 1);
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
$C_3^2:D_6$ (as 18T57):
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 108 |
| The 11 conjugacy class representatives for $C_3^2:D_6$ |
| Character table for $C_3^2:D_6$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.1.23.1, 6.0.14283.1, 9.1.239483061.1 x3 |
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 9 siblings: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 27 sibling: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 9.1.239483061.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{3}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{9}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $18$ | $6$ | $3$ | $21$ | |||
|
\(23\)
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |