Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 44*x^10 - 38*x^9 + 855*x^8 + 1626*x^7 - 7196*x^6 - 23416*x^5 + 21953*x^4 + 169301*x^3 + 261476*x^2 + 175353*x + 54317)
gp: K = bnfinit(y^12 - y^11 - 44*y^10 - 38*y^9 + 855*y^8 + 1626*y^7 - 7196*y^6 - 23416*y^5 + 21953*y^4 + 169301*y^3 + 261476*y^2 + 175353*y + 54317, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 - 44*x^10 - 38*x^9 + 855*x^8 + 1626*x^7 - 7196*x^6 - 23416*x^5 + 21953*x^4 + 169301*x^3 + 261476*x^2 + 175353*x + 54317);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 - 44*x^10 - 38*x^9 + 855*x^8 + 1626*x^7 - 7196*x^6 - 23416*x^5 + 21953*x^4 + 169301*x^3 + 261476*x^2 + 175353*x + 54317)
\( x^{12} - x^{11} - 44 x^{10} - 38 x^{9} + 855 x^{8} + 1626 x^{7} - 7196 x^{6} - 23416 x^{5} + \cdots + 54317 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(37248086166780626891533\)
\(\medspace = 13^{9}\cdot 37^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(76.02\) | 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: | $13^{3/4}37^{2/3}\approx 76.01955626985882$ | ||
| Ramified primes: |
\(13\), \(37\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{13}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(481=13\cdot 37\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{481}(1,·)$, $\chi_{481}(454,·)$, $\chi_{481}(38,·)$, $\chi_{481}(417,·)$, $\chi_{481}(359,·)$, $\chi_{481}(174,·)$, $\chi_{481}(47,·)$, $\chi_{481}(112,·)$, $\chi_{481}(408,·)$, $\chi_{481}(343,·)$, $\chi_{481}(248,·)$, $\chi_{481}(285,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | 4.0.2197.1$^{2}$, 12.0.37248086166780626891533.1$^{30}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{3}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{87}a^{10}+\frac{2}{29}a^{9}-\frac{2}{29}a^{8}+\frac{4}{29}a^{7}+\frac{2}{29}a^{6}-\frac{11}{29}a^{5}+\frac{31}{87}a^{4}-\frac{4}{29}a^{3}-\frac{5}{87}a^{2}+\frac{41}{87}a$, $\frac{1}{40\!\cdots\!93}a^{11}-\frac{16\!\cdots\!78}{40\!\cdots\!93}a^{10}+\frac{87\!\cdots\!55}{40\!\cdots\!93}a^{9}+\frac{37\!\cdots\!91}{13\!\cdots\!31}a^{8}-\frac{22\!\cdots\!15}{13\!\cdots\!31}a^{7}+\frac{42\!\cdots\!29}{13\!\cdots\!31}a^{6}-\frac{19\!\cdots\!89}{40\!\cdots\!93}a^{5}+\frac{15\!\cdots\!16}{40\!\cdots\!93}a^{4}+\frac{49\!\cdots\!77}{40\!\cdots\!93}a^{3}-\frac{15\!\cdots\!53}{40\!\cdots\!93}a^{2}+\frac{13\!\cdots\!30}{40\!\cdots\!93}a+\frac{59\!\cdots\!69}{14\!\cdots\!17}$
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
$C_{73}$, which has order $73$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $73$ (assuming GRH)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $5$ | 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: |
$\frac{1061468995}{154549481005949}a^{11}-\frac{277641725}{154549481005949}a^{10}-\frac{174637338421}{463648443017847}a^{9}-\frac{19699659512}{154549481005949}a^{8}+\frac{1164181131720}{154549481005949}a^{7}+\frac{1241417278236}{154549481005949}a^{6}-\frac{11940723213833}{154549481005949}a^{5}-\frac{14513831675787}{154549481005949}a^{4}+\frac{153846502651670}{463648443017847}a^{3}+\frac{125513291209767}{154549481005949}a^{2}+\frac{295197251693708}{463648443017847}a+\frac{8451427162898}{15987877345443}$, $\frac{26\!\cdots\!21}{13\!\cdots\!31}a^{11}-\frac{45\!\cdots\!01}{13\!\cdots\!31}a^{10}-\frac{11\!\cdots\!69}{13\!\cdots\!31}a^{9}-\frac{75\!\cdots\!61}{13\!\cdots\!31}a^{8}+\frac{23\!\cdots\!18}{13\!\cdots\!31}a^{7}+\frac{24\!\cdots\!42}{13\!\cdots\!31}a^{6}-\frac{22\!\cdots\!57}{13\!\cdots\!31}a^{5}-\frac{44\!\cdots\!99}{13\!\cdots\!31}a^{4}+\frac{10\!\cdots\!98}{13\!\cdots\!31}a^{3}+\frac{37\!\cdots\!28}{13\!\cdots\!31}a^{2}+\frac{31\!\cdots\!04}{13\!\cdots\!31}a+\frac{43\!\cdots\!37}{46\!\cdots\!39}$, $\frac{26\!\cdots\!21}{13\!\cdots\!31}a^{11}-\frac{45\!\cdots\!01}{13\!\cdots\!31}a^{10}-\frac{11\!\cdots\!69}{13\!\cdots\!31}a^{9}-\frac{75\!\cdots\!61}{13\!\cdots\!31}a^{8}+\frac{23\!\cdots\!18}{13\!\cdots\!31}a^{7}+\frac{24\!\cdots\!42}{13\!\cdots\!31}a^{6}-\frac{22\!\cdots\!57}{13\!\cdots\!31}a^{5}-\frac{44\!\cdots\!99}{13\!\cdots\!31}a^{4}+\frac{10\!\cdots\!98}{13\!\cdots\!31}a^{3}+\frac{37\!\cdots\!28}{13\!\cdots\!31}a^{2}+\frac{31\!\cdots\!04}{13\!\cdots\!31}a+\frac{38\!\cdots\!98}{46\!\cdots\!39}$, $\frac{57\!\cdots\!03}{40\!\cdots\!93}a^{11}-\frac{88\!\cdots\!95}{40\!\cdots\!93}a^{10}-\frac{25\!\cdots\!93}{40\!\cdots\!93}a^{9}-\frac{23\!\cdots\!73}{13\!\cdots\!31}a^{8}+\frac{59\!\cdots\!01}{46\!\cdots\!39}a^{7}+\frac{20\!\cdots\!68}{13\!\cdots\!31}a^{6}-\frac{48\!\cdots\!62}{40\!\cdots\!93}a^{5}-\frac{10\!\cdots\!25}{40\!\cdots\!93}a^{4}+\frac{23\!\cdots\!08}{40\!\cdots\!93}a^{3}+\frac{29\!\cdots\!68}{14\!\cdots\!17}a^{2}+\frac{74\!\cdots\!80}{40\!\cdots\!93}a+\frac{26\!\cdots\!28}{14\!\cdots\!17}$, $\frac{34\!\cdots\!88}{40\!\cdots\!93}a^{11}-\frac{21\!\cdots\!55}{13\!\cdots\!31}a^{10}-\frac{14\!\cdots\!20}{40\!\cdots\!93}a^{9}-\frac{16\!\cdots\!04}{13\!\cdots\!31}a^{8}+\frac{10\!\cdots\!71}{13\!\cdots\!31}a^{7}+\frac{10\!\cdots\!09}{13\!\cdots\!31}a^{6}-\frac{29\!\cdots\!22}{40\!\cdots\!93}a^{5}-\frac{19\!\cdots\!48}{13\!\cdots\!31}a^{4}+\frac{46\!\cdots\!98}{13\!\cdots\!31}a^{3}+\frac{49\!\cdots\!78}{40\!\cdots\!93}a^{2}+\frac{42\!\cdots\!41}{40\!\cdots\!93}a+\frac{16\!\cdots\!58}{14\!\cdots\!17}$
| 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: | \( 17287.4310274 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 17287.4310274 \cdot 73}{2\cdot\sqrt{37248086166780626891533}}\cr\approx \mathstrut & 0.201164134644 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 44*x^10 - 38*x^9 + 855*x^8 + 1626*x^7 - 7196*x^6 - 23416*x^5 + 21953*x^4 + 169301*x^3 + 261476*x^2 + 175353*x + 54317)
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^12 - x^11 - 44*x^10 - 38*x^9 + 855*x^8 + 1626*x^7 - 7196*x^6 - 23416*x^5 + 21953*x^4 + 169301*x^3 + 261476*x^2 + 175353*x + 54317, 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^12 - x^11 - 44*x^10 - 38*x^9 + 855*x^8 + 1626*x^7 - 7196*x^6 - 23416*x^5 + 21953*x^4 + 169301*x^3 + 261476*x^2 + 175353*x + 54317);
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^12 - x^11 - 44*x^10 - 38*x^9 + 855*x^8 + 1626*x^7 - 7196*x^6 - 23416*x^5 + 21953*x^4 + 169301*x^3 + 261476*x^2 + 175353*x + 54317);
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
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 cyclic group of order 12 |
| The 12 conjugacy class representatives for $C_{12}$ |
| Character table for $C_{12}$ |
Intermediate fields
| \(\Q(\sqrt{13}) \), 3.3.1369.1, 4.0.2197.1, 6.6.4117531717.1 |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.12.0.1}{12} }$ | ${\href{/padicField/3.3.0.1}{3} }^{4}$ | ${\href{/padicField/5.12.0.1}{12} }$ | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.12.0.1}{12} }$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.1.0.1}{1} }^{12}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.4.0.1}{4} }^{3}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.12.0.1}{12} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(13\)
| 13.12.9.2 | $x^{12} + 8 x^{10} + 44 x^{9} + 63 x^{8} + 264 x^{7} + 550 x^{6} - 6336 x^{5} + 3843 x^{4} + 4532 x^{3} + 46454 x^{2} + 30668 x + 30982$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
|
\(37\)
| 37.12.8.1 | $x^{12} + 18 x^{10} + 220 x^{9} + 114 x^{8} + 864 x^{7} - 5754 x^{6} + 7320 x^{5} - 47346 x^{4} - 240044 x^{3} + 340080 x^{2} - 2045220 x + 8612757$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |