Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 285*x^10 + 285*x^9 + 27236*x^8 - 33450*x^7 - 1015091*x^6 + 1882750*x^5 + 13705706*x^4 - 36747465*x^3 - 37348115*x^2 + 178599199*x - 130101919)
gp: K = bnfinit(y^12 - y^11 - 285*y^10 + 285*y^9 + 27236*y^8 - 33450*y^7 - 1015091*y^6 + 1882750*y^5 + 13705706*y^4 - 36747465*y^3 - 37348115*y^2 + 178599199*y - 130101919, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 - 285*x^10 + 285*x^9 + 27236*x^8 - 33450*x^7 - 1015091*x^6 + 1882750*x^5 + 13705706*x^4 - 36747465*x^3 - 37348115*x^2 + 178599199*x - 130101919);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 - 285*x^10 + 285*x^9 + 27236*x^8 - 33450*x^7 - 1015091*x^6 + 1882750*x^5 + 13705706*x^4 - 36747465*x^3 - 37348115*x^2 + 178599199*x - 130101919)
\( x^{12} - x^{11} - 285 x^{10} + 285 x^{9} + 27236 x^{8} - 33450 x^{7} - 1015091 x^{6} + 1882750 x^{5} + \cdots - 130101919 \)
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: | $[12, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(84489053066670461041015625\)
\(\medspace = 5^{9}\cdot 13^{11}\cdot 17^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(144.73\) | 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: | $5^{3/4}13^{11/12}17^{1/2}\approx 144.7327131886099$ | ||
| Ramified primes: |
\(5\), \(13\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{65}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1105=5\cdot 13\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1105}(256,·)$, $\chi_{1105}(1,·)$, $\chi_{1105}(324,·)$, $\chi_{1105}(69,·)$, $\chi_{1105}(1089,·)$, $\chi_{1105}(968,·)$, $\chi_{1105}(492,·)$, $\chi_{1105}(288,·)$, $\chi_{1105}(341,·)$, $\chi_{1105}(1087,·)$, $\chi_{1105}(798,·)$, $\chi_{1105}(917,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{83}a^{10}-\frac{41}{83}a^{9}-\frac{25}{83}a^{8}+\frac{14}{83}a^{7}+\frac{5}{83}a^{6}-\frac{16}{83}a^{5}-\frac{36}{83}a^{4}+\frac{9}{83}a^{3}+\frac{17}{83}a^{2}-\frac{31}{83}a$, $\frac{1}{55\!\cdots\!67}a^{11}-\frac{21\!\cdots\!26}{55\!\cdots\!67}a^{10}-\frac{19\!\cdots\!57}{55\!\cdots\!67}a^{9}+\frac{11\!\cdots\!83}{55\!\cdots\!67}a^{8}+\frac{16\!\cdots\!49}{55\!\cdots\!67}a^{7}-\frac{10\!\cdots\!98}{55\!\cdots\!67}a^{6}-\frac{95\!\cdots\!34}{55\!\cdots\!67}a^{5}+\frac{74\!\cdots\!45}{55\!\cdots\!67}a^{4}-\frac{15\!\cdots\!96}{55\!\cdots\!67}a^{3}+\frac{17\!\cdots\!79}{55\!\cdots\!67}a^{2}+\frac{16\!\cdots\!63}{55\!\cdots\!67}a+\frac{19\!\cdots\!23}{67\!\cdots\!49}$
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
$C_{2}\times C_{2}$, which has order $4$ (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: | $11$ | 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: |
$\frac{32\!\cdots\!45}{87\!\cdots\!69}a^{11}+\frac{79\!\cdots\!96}{87\!\cdots\!69}a^{10}-\frac{89\!\cdots\!45}{87\!\cdots\!69}a^{9}-\frac{21\!\cdots\!55}{87\!\cdots\!69}a^{8}+\frac{80\!\cdots\!30}{87\!\cdots\!69}a^{7}+\frac{17\!\cdots\!75}{87\!\cdots\!69}a^{6}-\frac{27\!\cdots\!03}{87\!\cdots\!69}a^{5}-\frac{33\!\cdots\!65}{87\!\cdots\!69}a^{4}+\frac{33\!\cdots\!40}{87\!\cdots\!69}a^{3}-\frac{16\!\cdots\!15}{87\!\cdots\!69}a^{2}-\frac{13\!\cdots\!75}{87\!\cdots\!69}a+\frac{15\!\cdots\!36}{10\!\cdots\!43}$, $\frac{10\!\cdots\!90}{87\!\cdots\!69}a^{11}+\frac{16\!\cdots\!64}{87\!\cdots\!69}a^{10}-\frac{30\!\cdots\!10}{87\!\cdots\!69}a^{9}-\frac{45\!\cdots\!80}{87\!\cdots\!69}a^{8}+\frac{28\!\cdots\!35}{87\!\cdots\!69}a^{7}+\frac{34\!\cdots\!35}{87\!\cdots\!69}a^{6}-\frac{10\!\cdots\!26}{87\!\cdots\!69}a^{5}-\frac{50\!\cdots\!80}{87\!\cdots\!69}a^{4}+\frac{13\!\cdots\!35}{87\!\cdots\!69}a^{3}-\frac{55\!\cdots\!45}{87\!\cdots\!69}a^{2}-\frac{56\!\cdots\!05}{87\!\cdots\!69}a+\frac{66\!\cdots\!16}{10\!\cdots\!43}$, $\frac{98\!\cdots\!56}{55\!\cdots\!67}a^{11}+\frac{13\!\cdots\!40}{55\!\cdots\!67}a^{10}-\frac{27\!\cdots\!78}{55\!\cdots\!67}a^{9}-\frac{37\!\cdots\!03}{55\!\cdots\!67}a^{8}+\frac{26\!\cdots\!09}{55\!\cdots\!67}a^{7}+\frac{28\!\cdots\!97}{55\!\cdots\!67}a^{6}-\frac{94\!\cdots\!06}{55\!\cdots\!67}a^{5}-\frac{36\!\cdots\!52}{55\!\cdots\!67}a^{4}+\frac{12\!\cdots\!33}{55\!\cdots\!67}a^{3}-\frac{57\!\cdots\!27}{55\!\cdots\!67}a^{2}-\frac{52\!\cdots\!60}{55\!\cdots\!67}a+\frac{61\!\cdots\!25}{67\!\cdots\!49}$, $\frac{17\!\cdots\!76}{55\!\cdots\!67}a^{11}+\frac{28\!\cdots\!94}{55\!\cdots\!67}a^{10}-\frac{48\!\cdots\!73}{55\!\cdots\!67}a^{9}-\frac{80\!\cdots\!80}{55\!\cdots\!67}a^{8}+\frac{44\!\cdots\!73}{55\!\cdots\!67}a^{7}+\frac{62\!\cdots\!31}{55\!\cdots\!67}a^{6}-\frac{15\!\cdots\!30}{55\!\cdots\!67}a^{5}-\frac{10\!\cdots\!52}{55\!\cdots\!67}a^{4}+\frac{20\!\cdots\!35}{55\!\cdots\!67}a^{3}-\frac{69\!\cdots\!47}{55\!\cdots\!67}a^{2}-\frac{84\!\cdots\!71}{55\!\cdots\!67}a+\frac{10\!\cdots\!90}{67\!\cdots\!49}$, $\frac{18\!\cdots\!06}{55\!\cdots\!67}a^{11}+\frac{42\!\cdots\!78}{55\!\cdots\!67}a^{10}-\frac{51\!\cdots\!69}{55\!\cdots\!67}a^{9}-\frac{11\!\cdots\!27}{55\!\cdots\!67}a^{8}+\frac{46\!\cdots\!38}{55\!\cdots\!67}a^{7}+\frac{93\!\cdots\!15}{55\!\cdots\!67}a^{6}-\frac{15\!\cdots\!71}{55\!\cdots\!67}a^{5}-\frac{18\!\cdots\!77}{55\!\cdots\!67}a^{4}+\frac{19\!\cdots\!03}{55\!\cdots\!67}a^{3}-\frac{18\!\cdots\!42}{55\!\cdots\!67}a^{2}-\frac{77\!\cdots\!23}{55\!\cdots\!67}a+\frac{91\!\cdots\!47}{67\!\cdots\!49}$, $\frac{25\!\cdots\!76}{57\!\cdots\!23}a^{11}+\frac{16\!\cdots\!97}{57\!\cdots\!23}a^{10}-\frac{71\!\cdots\!81}{57\!\cdots\!23}a^{9}-\frac{44\!\cdots\!34}{57\!\cdots\!23}a^{8}+\frac{64\!\cdots\!45}{57\!\cdots\!23}a^{7}+\frac{37\!\cdots\!58}{57\!\cdots\!23}a^{6}-\frac{23\!\cdots\!81}{57\!\cdots\!23}a^{5}-\frac{98\!\cdots\!33}{57\!\cdots\!23}a^{4}+\frac{43\!\cdots\!11}{57\!\cdots\!23}a^{3}+\frac{64\!\cdots\!07}{57\!\cdots\!23}a^{2}-\frac{36\!\cdots\!26}{57\!\cdots\!23}a+\frac{37\!\cdots\!66}{69\!\cdots\!81}$, $\frac{41\!\cdots\!47}{57\!\cdots\!23}a^{11}+\frac{62\!\cdots\!88}{57\!\cdots\!23}a^{10}-\frac{11\!\cdots\!96}{57\!\cdots\!23}a^{9}-\frac{17\!\cdots\!84}{57\!\cdots\!23}a^{8}+\frac{10\!\cdots\!14}{57\!\cdots\!23}a^{7}+\frac{13\!\cdots\!19}{57\!\cdots\!23}a^{6}-\frac{38\!\cdots\!41}{57\!\cdots\!23}a^{5}-\frac{18\!\cdots\!12}{57\!\cdots\!23}a^{4}+\frac{51\!\cdots\!38}{57\!\cdots\!23}a^{3}-\frac{23\!\cdots\!58}{57\!\cdots\!23}a^{2}-\frac{21\!\cdots\!39}{57\!\cdots\!23}a+\frac{26\!\cdots\!97}{69\!\cdots\!81}$, $\frac{85\!\cdots\!22}{55\!\cdots\!67}a^{11}+\frac{13\!\cdots\!71}{55\!\cdots\!67}a^{10}-\frac{23\!\cdots\!38}{55\!\cdots\!67}a^{9}-\frac{37\!\cdots\!64}{55\!\cdots\!67}a^{8}+\frac{21\!\cdots\!48}{55\!\cdots\!67}a^{7}+\frac{27\!\cdots\!53}{55\!\cdots\!67}a^{6}-\frac{77\!\cdots\!26}{55\!\cdots\!67}a^{5}-\frac{30\!\cdots\!83}{55\!\cdots\!67}a^{4}+\frac{10\!\cdots\!42}{55\!\cdots\!67}a^{3}-\frac{54\!\cdots\!10}{55\!\cdots\!67}a^{2}-\frac{42\!\cdots\!68}{55\!\cdots\!67}a+\frac{55\!\cdots\!15}{67\!\cdots\!49}$, $\frac{12\!\cdots\!58}{55\!\cdots\!67}a^{11}+\frac{23\!\cdots\!26}{55\!\cdots\!67}a^{10}-\frac{36\!\cdots\!62}{55\!\cdots\!67}a^{9}-\frac{47\!\cdots\!80}{55\!\cdots\!67}a^{8}+\frac{36\!\cdots\!58}{55\!\cdots\!67}a^{7}+\frac{19\!\cdots\!94}{55\!\cdots\!67}a^{6}-\frac{14\!\cdots\!87}{55\!\cdots\!67}a^{5}+\frac{81\!\cdots\!74}{55\!\cdots\!67}a^{4}+\frac{19\!\cdots\!85}{55\!\cdots\!67}a^{3}-\frac{33\!\cdots\!47}{55\!\cdots\!67}a^{2}-\frac{45\!\cdots\!60}{55\!\cdots\!67}a+\frac{11\!\cdots\!07}{67\!\cdots\!49}$, $\frac{81\!\cdots\!90}{55\!\cdots\!67}a^{11}+\frac{98\!\cdots\!40}{55\!\cdots\!67}a^{10}-\frac{23\!\cdots\!66}{55\!\cdots\!67}a^{9}-\frac{27\!\cdots\!01}{55\!\cdots\!67}a^{8}+\frac{21\!\cdots\!74}{55\!\cdots\!67}a^{7}+\frac{20\!\cdots\!72}{55\!\cdots\!67}a^{6}-\frac{78\!\cdots\!38}{55\!\cdots\!67}a^{5}-\frac{18\!\cdots\!00}{55\!\cdots\!67}a^{4}+\frac{10\!\cdots\!33}{55\!\cdots\!67}a^{3}-\frac{63\!\cdots\!98}{55\!\cdots\!67}a^{2}-\frac{44\!\cdots\!70}{55\!\cdots\!67}a+\frac{58\!\cdots\!07}{67\!\cdots\!49}$, $\frac{13\!\cdots\!59}{55\!\cdots\!67}a^{11}+\frac{14\!\cdots\!28}{55\!\cdots\!67}a^{10}-\frac{37\!\cdots\!39}{55\!\cdots\!67}a^{9}-\frac{41\!\cdots\!25}{55\!\cdots\!67}a^{8}+\frac{34\!\cdots\!12}{55\!\cdots\!67}a^{7}+\frac{31\!\cdots\!26}{55\!\cdots\!67}a^{6}-\frac{11\!\cdots\!88}{55\!\cdots\!67}a^{5}-\frac{30\!\cdots\!29}{55\!\cdots\!67}a^{4}+\frac{14\!\cdots\!29}{55\!\cdots\!67}a^{3}-\frac{72\!\cdots\!77}{55\!\cdots\!67}a^{2}-\frac{56\!\cdots\!63}{55\!\cdots\!67}a+\frac{70\!\cdots\!21}{67\!\cdots\!49}$
| 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: | \( 544261614.045 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 544261614.045 \cdot 4}{2\cdot\sqrt{84489053066670461041015625}}\cr\approx \mathstrut & 0.485062194837 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 285*x^10 + 285*x^9 + 27236*x^8 - 33450*x^7 - 1015091*x^6 + 1882750*x^5 + 13705706*x^4 - 36747465*x^3 - 37348115*x^2 + 178599199*x - 130101919)
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 - 285*x^10 + 285*x^9 + 27236*x^8 - 33450*x^7 - 1015091*x^6 + 1882750*x^5 + 13705706*x^4 - 36747465*x^3 - 37348115*x^2 + 178599199*x - 130101919, 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 - 285*x^10 + 285*x^9 + 27236*x^8 - 33450*x^7 - 1015091*x^6 + 1882750*x^5 + 13705706*x^4 - 36747465*x^3 - 37348115*x^2 + 178599199*x - 130101919);
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 - 285*x^10 + 285*x^9 + 27236*x^8 - 33450*x^7 - 1015091*x^6 + 1882750*x^5 + 13705706*x^4 - 36747465*x^3 - 37348115*x^2 + 178599199*x - 130101919);
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
| \(\Q(\sqrt{65}) \), 3.3.169.1, 4.4.79366625.1, 6.6.46411625.1 |
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 | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | ${\href{/padicField/3.12.0.1}{12} }$ | R | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.12.0.1}{12} }$ | R | R | ${\href{/padicField/19.12.0.1}{12} }$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.12.0.1}{12} }$ | ${\href{/padicField/47.1.0.1}{1} }^{12}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.12.0.1}{12} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.4.3.4 | $x^{4} + 15$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.4.3.4 | $x^{4} + 15$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.4 | $x^{4} + 15$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
|
\(13\)
| 13.12.11.8 | $x^{12} + 104$ | $12$ | $1$ | $11$ | $C_{12}$ | $[\ ]_{12}$ |
|
\(17\)
| 17.12.6.2 | $x^{12} + 578 x^{8} + 835210 x^{4} - 4259571 x^{2} + 72412707$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ |