Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^15 + 226*x^12 + 445*x^9 + 7252*x^6 + 17427*x^3 + 50653)
gp: K = bnfinit(y^18 - 7*y^15 + 226*y^12 + 445*y^9 + 7252*y^6 + 17427*y^3 + 50653, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 7*x^15 + 226*x^12 + 445*x^9 + 7252*x^6 + 17427*x^3 + 50653);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 7*x^15 + 226*x^12 + 445*x^9 + 7252*x^6 + 17427*x^3 + 50653)
\( x^{18} - 7x^{15} + 226x^{12} + 445x^{9} + 7252x^{6} + 17427x^{3} + 50653 \)
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: |
\(-150193475160708469172913491816448\)
\(\medspace = -\,2^{12}\cdot 3^{27}\cdot 37^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(61.32\) | 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: | $2^{2/3}3^{3/2}37^{5/6}\approx 167.1873122187818$ | ||
| Ramified primes: |
\(2\), \(3\), \(37\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{37}a^{12}-\frac{17}{37}a^{9}-\frac{11}{37}a^{6}$, $\frac{1}{37}a^{13}-\frac{17}{37}a^{10}-\frac{11}{37}a^{7}$, $\frac{1}{37}a^{14}-\frac{17}{37}a^{11}-\frac{11}{37}a^{8}$, $\frac{1}{75422052413}a^{15}-\frac{668060703}{75422052413}a^{12}+\frac{19764945802}{75422052413}a^{9}-\frac{23972877630}{75422052413}a^{6}-\frac{670894095}{2038433849}a^{3}-\frac{229409917}{2038433849}$, $\frac{1}{75422052413}a^{16}-\frac{668060703}{75422052413}a^{13}+\frac{19764945802}{75422052413}a^{10}-\frac{23972877630}{75422052413}a^{7}-\frac{670894095}{2038433849}a^{4}-\frac{229409917}{2038433849}a$, $\frac{1}{2790615939281}a^{17}+\frac{25831579334}{2790615939281}a^{14}-\frac{355306882414}{2790615939281}a^{11}-\frac{1220533546993}{2790615939281}a^{8}+\frac{36020915187}{75422052413}a^{5}+\frac{24231796271}{75422052413}a^{2}$
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
$C_{3}$, which has order $3$ (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: |
\( -\frac{74094}{2038433849} a^{15} + \frac{572815}{2038433849} a^{12} - \frac{17851714}{2038433849} a^{9} - \frac{15356702}{2038433849} a^{6} - \frac{768371008}{2038433849} a^{3} + \frac{168137658}{2038433849} \)
(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: |
$\frac{8340663}{75422052413}a^{15}-\frac{68855344}{75422052413}a^{12}+\frac{1957051835}{75422052413}a^{9}+\frac{1492470148}{75422052413}a^{6}+\frac{1165371066}{2038433849}a^{3}+\frac{2201956651}{2038433849}$, $\frac{19325}{280379377}a^{15}-\frac{70343}{280379377}a^{12}+\frac{3221759}{280379377}a^{9}+\frac{23668315}{280379377}a^{6}+\frac{1963803}{7577821}a^{3}+\frac{4485278}{7577821}$, $\frac{141734219}{2790615939281}a^{17}-\frac{74094}{2038433849}a^{15}-\frac{1607421894}{2790615939281}a^{14}+\frac{572815}{2038433849}a^{12}+\frac{36649178405}{2790615939281}a^{11}-\frac{17851714}{2038433849}a^{9}-\frac{63126139724}{2790615939281}a^{8}-\frac{15356702}{2038433849}a^{6}+\frac{19988833389}{75422052413}a^{5}-\frac{768371008}{2038433849}a^{3}+\frac{26751934077}{75422052413}a^{2}+\frac{2206571507}{2038433849}$, $\frac{130302243}{2790615939281}a^{17}+\frac{74094}{2038433849}a^{15}-\frac{1018993938}{2790615939281}a^{14}-\frac{572815}{2038433849}a^{12}+\frac{27639169385}{2790615939281}a^{11}+\frac{17851714}{2038433849}a^{9}+\frac{61800586005}{2790615939281}a^{8}+\frac{15356702}{2038433849}a^{6}+\frac{8902140233}{75422052413}a^{5}+\frac{768371008}{2038433849}a^{3}+\frac{67964470629}{75422052413}a^{2}-\frac{2206571507}{2038433849}$, $\frac{176273956}{2790615939281}a^{17}+\frac{13856881}{75422052413}a^{16}-\frac{18676030}{75422052413}a^{15}+\frac{510494705}{2790615939281}a^{14}-\frac{178958891}{75422052413}a^{13}+\frac{172554679}{75422052413}a^{12}+\frac{26770363058}{2790615939281}a^{11}+\frac{3489402258}{75422052413}a^{10}-\frac{3789827463}{75422052413}a^{9}+\frac{432596515491}{2790615939281}a^{8}-\frac{9058149634}{75422052413}a^{7}-\frac{6259718629}{75422052413}a^{6}+\frac{57354718007}{75422052413}a^{5}+\frac{119636346}{2038433849}a^{4}+\frac{982262248}{2038433849}a^{3}+\frac{174609955196}{75422052413}a^{2}-\frac{2985948810}{2038433849}a+\frac{2652573501}{2038433849}$, $\frac{33475}{280379377}a^{16}+\frac{13856881}{75422052413}a^{15}-\frac{121849}{280379377}a^{13}-\frac{178958891}{75422052413}a^{12}+\frac{5943486}{280379377}a^{10}+\frac{3489402258}{75422052413}a^{9}+\frac{44262986}{280379377}a^{7}-\frac{9058149634}{75422052413}a^{6}+\frac{3666408}{7577821}a^{4}+\frac{119636346}{2038433849}a^{3}+\frac{10808424}{7577821}a-\frac{2985948810}{2038433849}$, $\frac{10\!\cdots\!00}{2790615939281}a^{17}-\frac{9859933703070}{75422052413}a^{16}-\frac{553420285755785}{75422052413}a^{15}-\frac{85\!\cdots\!85}{2790615939281}a^{14}+\frac{10\!\cdots\!76}{75422052413}a^{13}+\frac{52\!\cdots\!62}{75422052413}a^{12}+\frac{23\!\cdots\!91}{2790615939281}a^{11}-\frac{10\!\cdots\!37}{75422052413}a^{10}-\frac{13\!\cdots\!40}{75422052413}a^{9}+\frac{25\!\cdots\!89}{2790615939281}a^{8}+\frac{24\!\cdots\!97}{75422052413}a^{7}-\frac{17\!\cdots\!35}{75422052413}a^{6}+\frac{15\!\cdots\!05}{75422052413}a^{5}-\frac{79\!\cdots\!72}{2038433849}a^{4}-\frac{88\!\cdots\!74}{2038433849}a^{3}+\frac{36\!\cdots\!39}{75422052413}a^{2}+\frac{15\!\cdots\!03}{2038433849}a-\frac{10\!\cdots\!52}{2038433849}$, $\frac{22\!\cdots\!22}{2790615939281}a^{17}+\frac{2333160358362}{75422052413}a^{16}-\frac{126568926146556}{75422052413}a^{15}-\frac{25\!\cdots\!38}{2790615939281}a^{14}-\frac{299644013272280}{75422052413}a^{13}+\frac{766369696858647}{75422052413}a^{12}+\frac{59\!\cdots\!50}{2790615939281}a^{11}+\frac{35\!\cdots\!79}{75422052413}a^{10}-\frac{25\!\cdots\!50}{75422052413}a^{9}-\frac{12\!\cdots\!56}{2790615939281}a^{8}-\frac{77\!\cdots\!78}{75422052413}a^{7}-\frac{97\!\cdots\!27}{75422052413}a^{6}+\frac{39\!\cdots\!74}{75422052413}a^{5}+\frac{26\!\cdots\!03}{2038433849}a^{4}-\frac{15\!\cdots\!27}{2038433849}a^{3}-\frac{38\!\cdots\!62}{75422052413}a^{2}-\frac{50\!\cdots\!49}{2038433849}a-\frac{87\!\cdots\!47}{2038433849}$
| 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: | \( 430430077.0591191 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 430430077.0591191 \cdot 3}{6\cdot\sqrt{150193475160708469172913491816448}}\cr\approx \mathstrut & 0.268019194313557 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^15 + 226*x^12 + 445*x^9 + 7252*x^6 + 17427*x^3 + 50653)
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 - 7*x^15 + 226*x^12 + 445*x^9 + 7252*x^6 + 17427*x^3 + 50653, 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 - 7*x^15 + 226*x^12 + 445*x^9 + 7252*x^6 + 17427*x^3 + 50653);
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 - 7*x^15 + 226*x^12 + 445*x^9 + 7252*x^6 + 17427*x^3 + 50653);
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
$C_3\times S_3^2$ (as 18T46):
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 27 conjugacy class representatives for $C_3\times S_3^2$ |
| Character table for $C_3\times S_3^2$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.3.148.1, 6.0.591408.1, 6.0.26946027.2 |
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
| Degree 12 sibling: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 27 sibling: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.0.348366465955974857566464.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | ${\href{/padicField/7.3.0.1}{3} }^{6}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | R | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{9}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.18.12.1 | $x^{18} + 10 x^{16} + 12 x^{15} + 28 x^{14} + 100 x^{13} + 92 x^{12} + 224 x^{11} + 640 x^{10} + 352 x^{9} + 736 x^{8} + 1760 x^{7} + 688 x^{6} + 1024 x^{5} + 1760 x^{4} + 448 x^{3} + 448 x^{2} + 320 x + 64$ | $3$ | $6$ | $12$ | $S_3 \times C_3$ | $[\ ]_{3}^{6}$ |
|
\(3\)
| Deg $18$ | $6$ | $3$ | $27$ | |||
|
\(37\)
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 37.2.1.1 | $x^{2} + 37$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 37.2.1.1 | $x^{2} + 37$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 37.2.1.1 | $x^{2} + 37$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 37.3.2.1 | $x^{3} + 37$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 37.6.5.1 | $x^{6} + 37$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |