Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 10*x^10 - 16*x^9 + 28*x^8 - 36*x^7 + 58*x^6 - 52*x^5 + 79*x^4 - 60*x^3 + 92*x^2 - 52*x + 41)
gp: K = bnfinit(y^12 - 4*y^11 + 10*y^10 - 16*y^9 + 28*y^8 - 36*y^7 + 58*y^6 - 52*y^5 + 79*y^4 - 60*y^3 + 92*y^2 - 52*y + 41, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 + 10*x^10 - 16*x^9 + 28*x^8 - 36*x^7 + 58*x^6 - 52*x^5 + 79*x^4 - 60*x^3 + 92*x^2 - 52*x + 41);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 + 10*x^10 - 16*x^9 + 28*x^8 - 36*x^7 + 58*x^6 - 52*x^5 + 79*x^4 - 60*x^3 + 92*x^2 - 52*x + 41)
\( x^{12} - 4 x^{11} + 10 x^{10} - 16 x^{9} + 28 x^{8} - 36 x^{7} + 58 x^{6} - 52 x^{5} + 79 x^{4} + \cdots + 41 \)
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: |
\(34162868224000000\)
\(\medspace = 2^{24}\cdot 5^{6}\cdot 19^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(23.87\) | 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: | $2^{2}5^{1/2}19^{2/3}\approx 63.686501757102$ | ||
| Ramified primes: |
\(2\), \(5\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3993007}a^{11}-\frac{369155}{3993007}a^{10}+\frac{594519}{3993007}a^{9}+\frac{360356}{3993007}a^{8}+\frac{1250477}{3993007}a^{7}+\frac{732179}{3993007}a^{6}-\frac{1959148}{3993007}a^{5}+\frac{29442}{3993007}a^{4}+\frac{421391}{3993007}a^{3}-\frac{1335402}{3993007}a^{2}+\frac{318595}{3993007}a+\frac{365281}{3993007}$
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
$C_{2}$, which has order $2$
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: | $5$ | 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: |
$\frac{550170}{3993007}a^{11}-\frac{1691309}{3993007}a^{10}+\frac{3342832}{3993007}a^{9}-\frac{3723044}{3993007}a^{8}+\frac{7776039}{3993007}a^{7}-\frac{7597758}{3993007}a^{6}+\frac{15840434}{3993007}a^{5}-\frac{5517266}{3993007}a^{4}+\frac{22665085}{3993007}a^{3}-\frac{8788382}{3993007}a^{2}+\frac{24340913}{3993007}a-\frac{5387547}{3993007}$, $\frac{286945}{3993007}a^{11}-\frac{691779}{3993007}a^{10}+\frac{1016394}{3993007}a^{9}-\frac{556852}{3993007}a^{8}+\frac{2520738}{3993007}a^{7}-\frac{953157}{3993007}a^{6}+\frac{3739663}{3993007}a^{5}+\frac{3024885}{3993007}a^{4}+\frac{7788535}{3993007}a^{3}+\frac{1989865}{3993007}a^{2}+\frac{3340017}{3993007}a+\frac{3115802}{3993007}$, $\frac{270776}{3993007}a^{11}-\frac{1370049}{3993007}a^{10}+\frac{3399539}{3993007}a^{9}-\frac{5348810}{3993007}a^{8}+\frac{8138580}{3993007}a^{7}-\frac{12268674}{3993007}a^{6}+\frac{16658165}{3993007}a^{5}-\frac{17820015}{3993007}a^{4}+\frac{18366419}{3993007}a^{3}-\frac{24035095}{3993007}a^{2}+\frac{18928520}{3993007}a-\frac{13427362}{3993007}$, $\frac{217406}{3993007}a^{11}-\frac{1064237}{3993007}a^{10}+\frac{2354131}{3993007}a^{9}-\frac{3233811}{3993007}a^{8}+\frac{5307081}{3993007}a^{7}-\frac{9102395}{3993007}a^{6}+\frac{11512616}{3993007}a^{5}-\frac{11901790}{3993007}a^{4}+\frac{13351166}{3993007}a^{3}-\frac{16826284}{3993007}a^{2}+\frac{13744169}{3993007}a-\frac{6628144}{3993007}$, $\frac{354281}{3993007}a^{11}-\frac{1644284}{3993007}a^{10}+\frac{3652603}{3993007}a^{9}-\frac{5121782}{3993007}a^{8}+\frac{8094408}{3993007}a^{7}-\frac{12584463}{3993007}a^{6}+\frac{15494222}{3993007}a^{5}-\frac{14965110}{3993007}a^{4}+\frac{16251183}{3993007}a^{3}-\frac{24072616}{3993007}a^{2}+\frac{13805347}{3993007}a-\frac{17210937}{3993007}$
| 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: | \( 597.4033513730107 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 597.4033513730107 \cdot 2}{2\cdot\sqrt{34162868224000000}}\cr\approx \mathstrut & 0.198870196058062 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 10*x^10 - 16*x^9 + 28*x^8 - 36*x^7 + 58*x^6 - 52*x^5 + 79*x^4 - 60*x^3 + 92*x^2 - 52*x + 41)
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 - 4*x^11 + 10*x^10 - 16*x^9 + 28*x^8 - 36*x^7 + 58*x^6 - 52*x^5 + 79*x^4 - 60*x^3 + 92*x^2 - 52*x + 41, 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 - 4*x^11 + 10*x^10 - 16*x^9 + 28*x^8 - 36*x^7 + 58*x^6 - 52*x^5 + 79*x^4 - 60*x^3 + 92*x^2 - 52*x + 41);
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 - 4*x^11 + 10*x^10 - 16*x^9 + 28*x^8 - 36*x^7 + 58*x^6 - 52*x^5 + 79*x^4 - 60*x^3 + 92*x^2 - 52*x + 41);
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
$C_6\times S_3$ (as 12T18):
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 solvable group of order 36 |
| The 18 conjugacy class representatives for $C_6\times S_3$ |
| Character table for $C_6\times S_3$ |
Intermediate fields
| \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{2}, \sqrt{-5})\), 6.0.23104000.1 |
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]
Sibling fields
| Galois closure: | deg 36 |
| Degree 18 siblings: | 18.6.297066099785564011102208000000.1, 18.0.4641657809149437673472000000000.1 |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{6}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.24.318 | $x^{12} + 10 x^{10} + 12 x^{9} + 110 x^{8} + 80 x^{7} + 752 x^{6} + 512 x^{5} + 1636 x^{4} + 1504 x^{3} + 1224 x^{2} + 1008 x - 648$ | $4$ | $3$ | $24$ | $C_6\times C_2$ | $[2, 3]^{3}$ |
|
\(5\)
| 5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
|
\(19\)
| 19.6.0.1 | $x^{6} + 17 x^{3} + 17 x^{2} + 6 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 19.6.4.1 | $x^{6} + 304 x^{3} - 5415$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.20.2t1.a.a | $1$ | $ 2^{2} \cdot 5 $ | \(\Q(\sqrt{-5}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.40.2t1.b.a | $1$ | $ 2^{3} \cdot 5 $ | \(\Q(\sqrt{-10}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.760.6t1.b.a | $1$ | $ 2^{3} \cdot 5 \cdot 19 $ | 6.0.8340544000.3 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.19.3t1.a.a | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.19.3t1.a.b | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.152.6t1.b.a | $1$ | $ 2^{3} \cdot 19 $ | 6.6.66724352.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.380.6t1.b.a | $1$ | $ 2^{2} \cdot 5 \cdot 19 $ | 6.0.1042568000.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.760.6t1.b.b | $1$ | $ 2^{3} \cdot 5 \cdot 19 $ | 6.0.8340544000.3 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.380.6t1.b.b | $1$ | $ 2^{2} \cdot 5 \cdot 19 $ | 6.0.1042568000.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.152.6t1.b.b | $1$ | $ 2^{3} \cdot 19 $ | 6.6.66724352.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 2.14440.3t2.a.a | $2$ | $ 2^{3} \cdot 5 \cdot 19^{2}$ | 3.1.14440.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.57760.6t3.a.a | $2$ | $ 2^{5} \cdot 5 \cdot 19^{2}$ | 6.0.16681088000.5 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| * | 2.760.6t5.a.a | $2$ | $ 2^{3} \cdot 5 \cdot 19 $ | 6.0.23104000.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.3040.12t18.c.a | $2$ | $ 2^{5} \cdot 5 \cdot 19 $ | 12.0.34162868224000000.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.760.6t5.a.b | $2$ | $ 2^{3} \cdot 5 \cdot 19 $ | 6.0.23104000.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.3040.12t18.c.b | $2$ | $ 2^{5} \cdot 5 \cdot 19 $ | 12.0.34162868224000000.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.