Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 40*x^10 + 72*x^9 + 482*x^8 - 782*x^7 - 1792*x^6 + 2008*x^5 + 2704*x^4 - 1192*x^3 - 1376*x^2 + 184*x + 208)
gp: K = bnfinit(y^12 - 2*y^11 - 40*y^10 + 72*y^9 + 482*y^8 - 782*y^7 - 1792*y^6 + 2008*y^5 + 2704*y^4 - 1192*y^3 - 1376*y^2 + 184*y + 208, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^11 - 40*x^10 + 72*x^9 + 482*x^8 - 782*x^7 - 1792*x^6 + 2008*x^5 + 2704*x^4 - 1192*x^3 - 1376*x^2 + 184*x + 208);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 2*x^11 - 40*x^10 + 72*x^9 + 482*x^8 - 782*x^7 - 1792*x^6 + 2008*x^5 + 2704*x^4 - 1192*x^3 - 1376*x^2 + 184*x + 208)
\( x^{12} - 2 x^{11} - 40 x^{10} + 72 x^{9} + 482 x^{8} - 782 x^{7} - 1792 x^{6} + 2008 x^{5} + 2704 x^{4} + \cdots + 208 \)
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: |
\(14515063671969809441792\)
\(\medspace = 2^{10}\cdot 13^{2}\cdot 17^{9}\cdot 29^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(70.28\) | 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: | $2^{37/30}13^{1/2}17^{3/4}29^{1/2}\approx 382.1885695819846$ | ||
| Ramified primes: |
\(2\), \(13\), \(17\), \(29\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{9}-\frac{1}{2}a^{4}$, $\frac{1}{12}a^{10}+\frac{1}{6}a^{6}+\frac{1}{6}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{1464165960}a^{11}+\frac{5631579}{244027660}a^{10}+\frac{5695707}{122013830}a^{9}+\frac{18267131}{244027660}a^{8}+\frac{162251749}{732082980}a^{7}+\frac{154617043}{732082980}a^{6}+\frac{45526753}{183020745}a^{5}+\frac{93155893}{366041490}a^{4}-\frac{121853491}{366041490}a^{3}-\frac{4650712}{183020745}a^{2}+\frac{28550986}{183020745}a-\frac{16667332}{61006915}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{2}$, which has order $2$ (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: |
$\frac{112393427}{1464165960}a^{11}-\frac{165645161}{732082980}a^{10}-\frac{177428728}{61006915}a^{9}+\frac{2041779467}{244027660}a^{8}+\frac{22801425413}{732082980}a^{7}-\frac{67693602689}{732082980}a^{6}-\frac{27242931523}{366041490}a^{5}+\frac{31141365027}{122013830}a^{4}+\frac{6213719411}{122013830}a^{3}-\frac{12470979993}{61006915}a^{2}-\frac{1080755706}{61006915}a+\frac{7697441423}{183020745}$, $\frac{93335417}{732082980}a^{11}-\frac{44200157}{122013830}a^{10}-\frac{587081097}{122013830}a^{9}+\frac{1616819357}{122013830}a^{8}+\frac{9327284074}{183020745}a^{7}-\frac{26241668867}{183020745}a^{6}-\frac{21143830073}{183020745}a^{5}+\frac{66168487961}{183020745}a^{4}+\frac{12085511458}{183020745}a^{3}-\frac{40786199263}{183020745}a^{2}-\frac{2581742381}{183020745}a+\frac{2327839607}{61006915}$, $\frac{52481369}{183020745}a^{11}-\frac{580582181}{732082980}a^{10}-\frac{2644876107}{244027660}a^{9}+\frac{1763897798}{61006915}a^{8}+\frac{21070612837}{183020745}a^{7}-\frac{18992153522}{61006915}a^{6}-\frac{16161341478}{61006915}a^{5}+\frac{280866317441}{366041490}a^{4}+\frac{28812961124}{183020745}a^{3}-\frac{80400199634}{183020745}a^{2}-\frac{5520127138}{183020745}a+\frac{12898174283}{183020745}$, $\frac{62039921}{1464165960}a^{11}-\frac{66505373}{732082980}a^{10}-\frac{407844611}{244027660}a^{9}+\frac{800137031}{244027660}a^{8}+\frac{14262954929}{732082980}a^{7}-\frac{25846757837}{732082980}a^{6}-\frac{11864015597}{183020745}a^{5}+\frac{5225738548}{61006915}a^{4}+\frac{9922916443}{122013830}a^{3}-\frac{2453895884}{61006915}a^{2}-\frac{1574565628}{61006915}a+\frac{423849989}{183020745}$, $\frac{41299}{1967965}a^{11}-\frac{779971}{7871860}a^{10}-\frac{5861921}{7871860}a^{9}+\frac{15075663}{3935930}a^{8}+\frac{13302612}{1967965}a^{7}-\frac{177450207}{3935930}a^{6}-\frac{3028549}{1967965}a^{5}+\frac{608835251}{3935930}a^{4}-\frac{60486406}{1967965}a^{3}-\frac{358656209}{1967965}a^{2}+\frac{6155382}{1967965}a+\frac{80533483}{1967965}$, $\frac{258111601}{732082980}a^{11}-\frac{545794411}{732082980}a^{10}-\frac{845289023}{61006915}a^{9}+\frac{3252259541}{122013830}a^{8}+\frac{29370827087}{183020745}a^{7}-\frac{17253945577}{61006915}a^{6}-\frac{63948912861}{122013830}a^{5}+\frac{118856726138}{183020745}a^{4}+\frac{113811953749}{183020745}a^{3}-\frac{42653023834}{183020745}a^{2}-\frac{25180886738}{183020745}a+\frac{4549177933}{183020745}$, $\frac{35660849}{183020745}a^{11}-\frac{96631279}{183020745}a^{10}-\frac{452941308}{61006915}a^{9}+\frac{2355536421}{122013830}a^{8}+\frac{29439077939}{366041490}a^{7}-\frac{38373459991}{183020745}a^{6}-\frac{37152527824}{183020745}a^{5}+\frac{32778052686}{61006915}a^{4}+\frac{9574771818}{61006915}a^{3}-\frac{21197280298}{61006915}a^{2}-\frac{2368053021}{61006915}a+\frac{11576024833}{183020745}$, $\frac{14318367}{48805532}a^{11}-\frac{26910655}{36604149}a^{10}-\frac{276202263}{24402766}a^{9}+\frac{326568737}{12201383}a^{8}+\frac{1538677710}{12201383}a^{7}-\frac{21248959885}{73208298}a^{6}-\frac{13116015701}{36604149}a^{5}+\frac{27065623297}{36604149}a^{4}+\frac{12801521081}{36604149}a^{3}-\frac{16800476603}{36604149}a^{2}-\frac{3234884836}{36604149}a+\frac{2782636603}{36604149}$, $\frac{51245687}{1464165960}a^{11}-\frac{48649093}{366041490}a^{10}-\frac{75428283}{61006915}a^{9}+\frac{1197544387}{244027660}a^{8}+\frac{8038094753}{732082980}a^{7}-\frac{38804379719}{732082980}a^{6}-\frac{135721049}{183020745}a^{5}+\frac{16318132087}{122013830}a^{4}-\frac{5237638679}{122013830}a^{3}-\frac{6137501513}{61006915}a^{2}+\frac{820170839}{61006915}a+\frac{3661928603}{183020745}$, $\frac{7582423}{1464165960}a^{11}-\frac{9792229}{732082980}a^{10}-\frac{23425759}{122013830}a^{9}+\frac{112546883}{244027660}a^{8}+\frac{1436140177}{732082980}a^{7}-\frac{3187100671}{732082980}a^{6}-\frac{1555079417}{366041490}a^{5}+\frac{545642483}{122013830}a^{4}+\frac{1061559619}{122013830}a^{3}+\frac{155492283}{61006915}a^{2}-\frac{226443944}{61006915}a-\frac{331287623}{183020745}$, $\frac{17214687}{488055320}a^{11}-\frac{39108589}{366041490}a^{10}-\frac{159465743}{122013830}a^{9}+\frac{954712861}{244027660}a^{8}+\frac{3225362723}{244027660}a^{7}-\frac{30973308607}{732082980}a^{6}-\frac{4111224337}{183020745}a^{5}+\frac{39206571853}{366041490}a^{4}-\frac{2473766701}{366041490}a^{3}-\frac{13108643587}{183020745}a^{2}+\frac{1142645296}{183020745}a+\frac{2680493939}{183020745}$
| 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: | \( 68111868.6417 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 68111868.6417 \cdot 2}{2\cdot\sqrt{14515063671969809441792}}\cr\approx \mathstrut & 2.31565187811 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 40*x^10 + 72*x^9 + 482*x^8 - 782*x^7 - 1792*x^6 + 2008*x^5 + 2704*x^4 - 1192*x^3 - 1376*x^2 + 184*x + 208)
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 - 2*x^11 - 40*x^10 + 72*x^9 + 482*x^8 - 782*x^7 - 1792*x^6 + 2008*x^5 + 2704*x^4 - 1192*x^3 - 1376*x^2 + 184*x + 208, 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 - 2*x^11 - 40*x^10 + 72*x^9 + 482*x^8 - 782*x^7 - 1792*x^6 + 2008*x^5 + 2704*x^4 - 1192*x^3 - 1376*x^2 + 184*x + 208);
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 - 2*x^11 - 40*x^10 + 72*x^9 + 482*x^8 - 782*x^7 - 1792*x^6 + 2008*x^5 + 2704*x^4 - 1192*x^3 - 1376*x^2 + 184*x + 208);
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
$A_5^2:C_4$ (as 12T278):
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 non-solvable group of order 14400 |
| The 22 conjugacy class representatives for $A_5^2:C_4$ |
| Character table for $A_5^2:C_4$ |
Intermediate fields
| \(\Q(\sqrt{17}) \) |
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 10 sibling: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 24 sibling: | data not computed |
| Degree 25 sibling: | data not computed |
| Degree 30 sibling: | data not computed |
| Degree 36 sibling: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 10.10.738604909015357696.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | R | R | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 2.5.4.1 | $x^{5} + 2$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ | |
| 2.6.6.8 | $x^{6} + 2 x + 2$ | $6$ | $1$ | $6$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ | |
|
\(13\)
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.5.0.1 | $x^{5} + 4 x + 11$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
|
\(17\)
| 17.12.9.1 | $x^{12} + 4 x^{10} + 56 x^{9} + 57 x^{8} + 168 x^{7} + 1044 x^{6} - 11256 x^{5} + 3356 x^{4} + 10080 x^{3} + 97736 x^{2} + 58576 x + 57252$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
|
\(29\)
| 29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 29.8.4.1 | $x^{8} + 2784 x^{7} + 2906616 x^{6} + 1348864734 x^{5} + 234834277018 x^{4} + 41857830864 x^{3} + 492109772617 x^{2} + 3561769809750 x + 616658760166$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |