Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 53*x^10 - 11*x^9 + 1303*x^8 + 438*x^7 + 18804*x^6 + 9784*x^5 + 170686*x^4 + 49088*x^3 + 900838*x^2 + 99987*x + 2139101)
gp: K = bnfinit(y^12 - y^11 + 53*y^10 - 11*y^9 + 1303*y^8 + 438*y^7 + 18804*y^6 + 9784*y^5 + 170686*y^4 + 49088*y^3 + 900838*y^2 + 99987*y + 2139101, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 + 53*x^10 - 11*x^9 + 1303*x^8 + 438*x^7 + 18804*x^6 + 9784*x^5 + 170686*x^4 + 49088*x^3 + 900838*x^2 + 99987*x + 2139101);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 + 53*x^10 - 11*x^9 + 1303*x^8 + 438*x^7 + 18804*x^6 + 9784*x^5 + 170686*x^4 + 49088*x^3 + 900838*x^2 + 99987*x + 2139101)
\( x^{12} - x^{11} + 53 x^{10} - 11 x^{9} + 1303 x^{8} + 438 x^{7} + 18804 x^{6} + 9784 x^{5} + \cdots + 2139101 \)
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: |
\(271773988103064453125\)
\(\medspace = 5^{9}\cdot 7^{8}\cdot 17^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(50.45\) | 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: | $5^{3/4}7^{2/3}17^{1/2}\approx 50.448778734178404$ | ||
| Ramified primes: |
\(5\), \(7\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(595=5\cdot 7\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{595}(256,·)$, $\chi_{595}(1,·)$, $\chi_{595}(67,·)$, $\chi_{595}(324,·)$, $\chi_{595}(492,·)$, $\chi_{595}(494,·)$, $\chi_{595}(288,·)$, $\chi_{595}(373,·)$, $\chi_{595}(86,·)$, $\chi_{595}(407,·)$, $\chi_{595}(239,·)$, $\chi_{595}(543,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | 4.0.36125.1$^{2}$, 12.0.271773988103064453125.1$^{30}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{29}a^{7}-\frac{4}{29}a^{6}+\frac{4}{29}a^{5}-\frac{10}{29}a^{4}+\frac{7}{29}a^{3}-\frac{11}{29}a^{2}+\frac{8}{29}a+\frac{5}{29}$, $\frac{1}{29}a^{8}-\frac{12}{29}a^{6}+\frac{6}{29}a^{5}-\frac{4}{29}a^{4}-\frac{12}{29}a^{3}-\frac{7}{29}a^{2}+\frac{8}{29}a-\frac{9}{29}$, $\frac{1}{29}a^{9}-\frac{13}{29}a^{6}-\frac{14}{29}a^{5}+\frac{13}{29}a^{4}-\frac{10}{29}a^{3}-\frac{8}{29}a^{2}+\frac{2}{29}$, $\frac{1}{29}a^{10}-\frac{8}{29}a^{6}+\frac{7}{29}a^{5}+\frac{5}{29}a^{4}-\frac{4}{29}a^{3}+\frac{2}{29}a^{2}-\frac{10}{29}a+\frac{7}{29}$, $\frac{1}{11\!\cdots\!39}a^{11}-\frac{48\!\cdots\!15}{40\!\cdots\!91}a^{10}+\frac{46\!\cdots\!12}{11\!\cdots\!39}a^{9}+\frac{15\!\cdots\!61}{11\!\cdots\!39}a^{8}+\frac{23\!\cdots\!12}{11\!\cdots\!39}a^{7}-\frac{39\!\cdots\!37}{11\!\cdots\!39}a^{6}+\frac{12\!\cdots\!33}{11\!\cdots\!39}a^{5}-\frac{60\!\cdots\!39}{11\!\cdots\!39}a^{4}+\frac{43\!\cdots\!32}{40\!\cdots\!91}a^{3}+\frac{19\!\cdots\!57}{11\!\cdots\!39}a^{2}-\frac{11\!\cdots\!82}{11\!\cdots\!39}a+\frac{15\!\cdots\!28}{11\!\cdots\!39}$
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
$C_{962}$, which has order $962$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $962$ (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: |
$\frac{1269530073}{544645043469941}a^{11}-\frac{97769500}{18780863567929}a^{10}+\frac{48823727828}{544645043469941}a^{9}-\frac{66433233759}{544645043469941}a^{8}+\frac{833247527752}{544645043469941}a^{7}-\frac{982514772374}{544645043469941}a^{6}+\frac{7605874835415}{544645043469941}a^{5}-\frac{14345995173038}{544645043469941}a^{4}+\frac{1522663426065}{18780863567929}a^{3}-\frac{104166015496291}{544645043469941}a^{2}+\frac{114126540599333}{544645043469941}a+\frac{25223255223958}{544645043469941}$, $\frac{57\!\cdots\!40}{11\!\cdots\!39}a^{11}+\frac{29\!\cdots\!24}{40\!\cdots\!91}a^{10}+\frac{38\!\cdots\!00}{11\!\cdots\!39}a^{9}+\frac{35\!\cdots\!95}{11\!\cdots\!39}a^{8}+\frac{14\!\cdots\!60}{11\!\cdots\!39}a^{7}+\frac{72\!\cdots\!50}{11\!\cdots\!39}a^{6}+\frac{27\!\cdots\!62}{11\!\cdots\!39}a^{5}+\frac{86\!\cdots\!35}{11\!\cdots\!39}a^{4}+\frac{95\!\cdots\!50}{40\!\cdots\!91}a^{3}+\frac{52\!\cdots\!75}{11\!\cdots\!39}a^{2}+\frac{93\!\cdots\!25}{11\!\cdots\!39}a+\frac{40\!\cdots\!16}{11\!\cdots\!39}$, $\frac{43\!\cdots\!65}{11\!\cdots\!39}a^{11}+\frac{13\!\cdots\!46}{40\!\cdots\!91}a^{10}+\frac{20\!\cdots\!00}{11\!\cdots\!39}a^{9}+\frac{43\!\cdots\!20}{11\!\cdots\!39}a^{8}+\frac{42\!\cdots\!90}{11\!\cdots\!39}a^{7}+\frac{13\!\cdots\!10}{11\!\cdots\!39}a^{6}+\frac{56\!\cdots\!27}{11\!\cdots\!39}a^{5}+\frac{17\!\cdots\!70}{11\!\cdots\!39}a^{4}+\frac{14\!\cdots\!35}{40\!\cdots\!91}a^{3}+\frac{11\!\cdots\!00}{11\!\cdots\!39}a^{2}+\frac{13\!\cdots\!25}{11\!\cdots\!39}a+\frac{39\!\cdots\!11}{11\!\cdots\!39}$, $\frac{71\!\cdots\!32}{11\!\cdots\!39}a^{11}-\frac{82\!\cdots\!54}{40\!\cdots\!91}a^{10}+\frac{30\!\cdots\!12}{11\!\cdots\!39}a^{9}+\frac{28\!\cdots\!59}{11\!\cdots\!39}a^{8}+\frac{61\!\cdots\!98}{11\!\cdots\!39}a^{7}+\frac{10\!\cdots\!64}{11\!\cdots\!39}a^{6}+\frac{73\!\cdots\!12}{11\!\cdots\!39}a^{5}+\frac{14\!\cdots\!68}{11\!\cdots\!39}a^{4}+\frac{18\!\cdots\!70}{40\!\cdots\!91}a^{3}+\frac{91\!\cdots\!11}{11\!\cdots\!39}a^{2}+\frac{15\!\cdots\!32}{11\!\cdots\!39}a+\frac{52\!\cdots\!32}{11\!\cdots\!39}$, $\frac{77\!\cdots\!72}{11\!\cdots\!39}a^{11}+\frac{21\!\cdots\!70}{40\!\cdots\!91}a^{10}+\frac{34\!\cdots\!12}{11\!\cdots\!39}a^{9}+\frac{63\!\cdots\!54}{11\!\cdots\!39}a^{8}+\frac{75\!\cdots\!58}{11\!\cdots\!39}a^{7}+\frac{18\!\cdots\!14}{11\!\cdots\!39}a^{6}+\frac{10\!\cdots\!74}{11\!\cdots\!39}a^{5}+\frac{23\!\cdots\!03}{11\!\cdots\!39}a^{4}+\frac{27\!\cdots\!20}{40\!\cdots\!91}a^{3}+\frac{14\!\cdots\!86}{11\!\cdots\!39}a^{2}+\frac{24\!\cdots\!57}{11\!\cdots\!39}a+\frac{56\!\cdots\!48}{11\!\cdots\!39}$
| 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: | \( 104.882003477 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 104.882003477 \cdot 962}{2\cdot\sqrt{271773988103064453125}}\cr\approx \mathstrut & 0.188287425628 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 53*x^10 - 11*x^9 + 1303*x^8 + 438*x^7 + 18804*x^6 + 9784*x^5 + 170686*x^4 + 49088*x^3 + 900838*x^2 + 99987*x + 2139101)
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 + 53*x^10 - 11*x^9 + 1303*x^8 + 438*x^7 + 18804*x^6 + 9784*x^5 + 170686*x^4 + 49088*x^3 + 900838*x^2 + 99987*x + 2139101, 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 + 53*x^10 - 11*x^9 + 1303*x^8 + 438*x^7 + 18804*x^6 + 9784*x^5 + 170686*x^4 + 49088*x^3 + 900838*x^2 + 99987*x + 2139101);
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 + 53*x^10 - 11*x^9 + 1303*x^8 + 438*x^7 + 18804*x^6 + 9784*x^5 + 170686*x^4 + 49088*x^3 + 900838*x^2 + 99987*x + 2139101);
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
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 4.0.36125.1, 6.6.300125.1 |
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 | ${\href{/padicField/2.12.0.1}{12} }$ | ${\href{/padicField/3.12.0.1}{12} }$ | R | R | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.1.0.1}{1} }^{12}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.12.0.1}{12} }$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.12.9.1 | $x^{12} - 30 x^{8} + 225 x^{4} + 1125$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
|
\(7\)
| 7.12.8.1 | $x^{12} + 15 x^{10} + 40 x^{9} + 84 x^{8} + 120 x^{7} + 53 x^{6} + 414 x^{5} - 1293 x^{4} - 1830 x^{3} + 10968 x^{2} - 13836 x + 12004$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |
|
\(17\)
| 17.12.6.2 | $x^{12} + 578 x^{8} + 835210 x^{4} - 4259571 x^{2} + 72412707$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ |