Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 76*x^27 + 5777*x^18 + 76*x^9 + 1)
gp: K = bnfinit(y^36 - 76*y^27 + 5777*y^18 + 76*y^9 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - 76*x^27 + 5777*x^18 + 76*x^9 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - 76*x^27 + 5777*x^18 + 76*x^9 + 1)
\( x^{36} - 76x^{27} + 5777x^{18} + 76x^{9} + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $36$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 18]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(33294538757658815101209249418169888839879795074462890625\)
\(\medspace = 3^{90}\cdot 5^{18}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(34.86\) | 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: | $3^{5/2}5^{1/2}\approx 34.85685011586675$ | ||
| Ramified primes: |
\(3\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $36$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(135=3^{3}\cdot 5\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{135}(1,·)$, $\chi_{135}(131,·)$, $\chi_{135}(4,·)$, $\chi_{135}(134,·)$, $\chi_{135}(11,·)$, $\chi_{135}(14,·)$, $\chi_{135}(16,·)$, $\chi_{135}(19,·)$, $\chi_{135}(26,·)$, $\chi_{135}(29,·)$, $\chi_{135}(31,·)$, $\chi_{135}(34,·)$, $\chi_{135}(41,·)$, $\chi_{135}(44,·)$, $\chi_{135}(46,·)$, $\chi_{135}(49,·)$, $\chi_{135}(56,·)$, $\chi_{135}(59,·)$, $\chi_{135}(61,·)$, $\chi_{135}(64,·)$, $\chi_{135}(71,·)$, $\chi_{135}(74,·)$, $\chi_{135}(76,·)$, $\chi_{135}(79,·)$, $\chi_{135}(86,·)$, $\chi_{135}(89,·)$, $\chi_{135}(91,·)$, $\chi_{135}(94,·)$, $\chi_{135}(101,·)$, $\chi_{135}(104,·)$, $\chi_{135}(106,·)$, $\chi_{135}(109,·)$, $\chi_{135}(116,·)$, $\chi_{135}(119,·)$, $\chi_{135}(121,·)$, $\chi_{135}(124,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{131072}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{34}a^{18}+\frac{13}{34}a^{9}-\frac{1}{34}$, $\frac{1}{34}a^{19}+\frac{13}{34}a^{10}-\frac{1}{34}a$, $\frac{1}{34}a^{20}+\frac{13}{34}a^{11}-\frac{1}{34}a^{2}$, $\frac{1}{34}a^{21}+\frac{13}{34}a^{12}-\frac{1}{34}a^{3}$, $\frac{1}{34}a^{22}+\frac{13}{34}a^{13}-\frac{1}{34}a^{4}$, $\frac{1}{34}a^{23}+\frac{13}{34}a^{14}-\frac{1}{34}a^{5}$, $\frac{1}{34}a^{24}+\frac{13}{34}a^{15}-\frac{1}{34}a^{6}$, $\frac{1}{34}a^{25}+\frac{13}{34}a^{16}-\frac{1}{34}a^{7}$, $\frac{1}{34}a^{26}+\frac{13}{34}a^{17}-\frac{1}{34}a^{8}$, $\frac{1}{196418}a^{27}-\frac{75025}{196418}$, $\frac{1}{196418}a^{28}-\frac{75025}{196418}a$, $\frac{1}{196418}a^{29}-\frac{75025}{196418}a^{2}$, $\frac{1}{196418}a^{30}-\frac{75025}{196418}a^{3}$, $\frac{1}{196418}a^{31}-\frac{75025}{196418}a^{4}$, $\frac{1}{196418}a^{32}-\frac{75025}{196418}a^{5}$, $\frac{1}{196418}a^{33}-\frac{75025}{196418}a^{6}$, $\frac{1}{196418}a^{34}-\frac{75025}{196418}a^{7}$, $\frac{1}{196418}a^{35}-\frac{75025}{196418}a^{8}$
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
$C_{37}$, which has order $37$ (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: | $17$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{21}{196418} a^{35} - \frac{9227465}{196418} a^{8} \)
(order $54$)
| 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: |
$\frac{3382}{98209}a^{30}-\frac{89}{34}a^{21}+\frac{6765}{34}a^{12}-\frac{89}{196418}a^{3}$, $\frac{4}{98209}a^{33}-\frac{5472}{98209}a^{30}+\frac{72}{17}a^{21}-\frac{5473}{17}a^{12}+\frac{1762289}{98209}a^{6}+\frac{72}{98209}a^{3}-1$, $\frac{21}{196418}a^{35}-\frac{5}{196418}a^{32}+\frac{1}{196418}a^{29}+\frac{9227465}{196418}a^{8}-\frac{2178309}{196418}a^{5}+\frac{514229}{196418}a^{2}$, $\frac{17711}{196418}a^{31}-\frac{233}{34}a^{22}+\frac{17711}{34}a^{13}+\frac{673018}{98209}a^{4}-1$, $\frac{60686}{98209}a^{35}-\frac{1597}{34}a^{26}+\frac{121393}{34}a^{17}-\frac{1597}{196418}a^{8}+1$, $\frac{1}{196418}a^{29}+\frac{1}{196418}a^{28}+\frac{514229}{196418}a^{2}+\frac{317811}{196418}a$, $\frac{121393}{196418}a^{35}-\frac{8854}{98209}a^{31}-\frac{1597}{34}a^{26}+\frac{233}{34}a^{22}+\frac{121393}{34}a^{17}-\frac{17711}{34}a^{13}+\frac{4612934}{98209}a^{8}+\frac{233}{196418}a^{4}$, $\frac{1}{196418}a^{29}+\frac{514229}{196418}a^{2}+1$, $\frac{17711}{196418}a^{31}-\frac{3382}{98209}a^{29}-\frac{233}{34}a^{22}+\frac{89}{34}a^{20}+\frac{17711}{34}a^{13}-\frac{6765}{34}a^{11}+\frac{673018}{98209}a^{4}+\frac{89}{196418}a^{2}$, $\frac{21}{196418}a^{35}-\frac{28657}{196418}a^{32}-\frac{1}{196418}a^{29}+\frac{377}{34}a^{23}-\frac{28657}{34}a^{14}+\frac{9227465}{196418}a^{8}-\frac{1088966}{98209}a^{5}-\frac{317811}{196418}a^{2}$, $\frac{21}{196418}a^{35}-\frac{5}{196418}a^{31}+\frac{9227465}{196418}a^{8}-\frac{2178309}{196418}a^{4}+1$, $\frac{13}{196418}a^{34}-\frac{23184}{98209}a^{33}+\frac{1}{196418}a^{29}+\frac{305}{17}a^{24}-\frac{23184}{17}a^{15}+\frac{5702887}{196418}a^{7}-\frac{1761984}{98209}a^{6}+\frac{317811}{196418}a^{2}$, $\frac{5776}{5777}a^{35}+\frac{23180}{98209}a^{33}-\frac{4181}{196418}a^{28}-76a^{26}-\frac{305}{17}a^{24}+\frac{55}{34}a^{19}+5777a^{17}+\frac{23184}{17}a^{15}-\frac{4181}{34}a^{10}-\frac{76}{5777}a^{8}-\frac{305}{98209}a^{6}-\frac{158878}{98209}a$, $\frac{4}{98209}a^{34}-\frac{14326}{98209}a^{32}+\frac{377}{34}a^{23}-\frac{28657}{34}a^{14}+\frac{1762289}{98209}a^{7}+\frac{377}{196418}a^{5}-1$, $\frac{4}{98209}a^{34}-\frac{28657}{196418}a^{32}-\frac{76}{5777}a^{27}+\frac{377}{34}a^{23}+a^{18}-\frac{28657}{34}a^{14}-76a^{9}+\frac{1762289}{98209}a^{7}-\frac{1088966}{98209}a^{5}-\frac{5776}{5777}$, $\frac{37506}{98209}a^{34}-\frac{23184}{98209}a^{33}+\frac{4181}{196418}a^{29}-\frac{987}{34}a^{25}+\frac{305}{17}a^{24}-\frac{55}{34}a^{20}+\frac{75025}{34}a^{16}-\frac{23184}{17}a^{15}+\frac{4181}{34}a^{11}-\frac{987}{196418}a^{7}-\frac{1761984}{98209}a^{6}+\frac{158878}{98209}a^{2}$, $\frac{5}{196418}a^{32}+\frac{3}{196418}a^{31}+\frac{1597}{196418}a^{27}-\frac{21}{34}a^{18}+\frac{1597}{34}a^{9}+\frac{2178309}{196418}a^{5}+\frac{1346269}{196418}a^{4}+\frac{60686}{98209}$
| 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: | \( 6550249244897.024 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 6550249244897.024 \cdot 37}{54\cdot\sqrt{33294538757658815101209249418169888839879795074462890625}}\cr\approx \mathstrut & 0.181182779997438 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^36 - 76*x^27 + 5777*x^18 + 76*x^9 + 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^36 - 76*x^27 + 5777*x^18 + 76*x^9 + 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^36 - 76*x^27 + 5777*x^18 + 76*x^9 + 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^36 - 76*x^27 + 5777*x^18 + 76*x^9 + 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
$C_2\times C_{18}$ (as 36T2):
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)
| An abelian group of order 36 |
| The 36 conjugacy class representatives for $C_2\times C_{18}$ |
| Character table for $C_2\times C_{18}$ |
Intermediate fields
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $18^{2}$ | R | R | $18^{2}$ | $18^{2}$ | $18^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{6}$ | ${\href{/padicField/19.3.0.1}{3} }^{12}$ | $18^{2}$ | $18^{2}$ | ${\href{/padicField/31.9.0.1}{9} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }^{6}$ | $18^{2}$ | $18^{2}$ | $18^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{18}$ | $18^{2}$ |
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 $36$ | $18$ | $2$ | $90$ | |||
|
\(5\)
| Deg $36$ | $2$ | $18$ | $18$ |