Normalized defining polynomial
\( x^{12} + 1095x^{10} + 160965x^{8} + 9460800x^{6} + 262241550x^{4} + 3368192625x^{2} + 15971752125 \)
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: |
\(1829654030234673819254664000000000\)
\(\medspace = 2^{12}\cdot 3^{6}\cdot 5^{9}\cdot 73^{11}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(591.38\) | 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: | $2\cdot 3^{1/2}5^{3/4}73^{11/12}\approx 591.3765255450226$ | ||
Ramified primes: |
\(2\), \(3\), \(5\), \(73\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{365}) \) | ||
$\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(4380=2^{2}\cdot 3\cdot 5\cdot 73\) | ||
Dirichlet character group: | $\lbrace$$\chi_{4380}(1,·)$, $\chi_{4380}(3649,·)$, $\chi_{4380}(649,·)$, $\chi_{4380}(1703,·)$, $\chi_{4380}(1487,·)$, $\chi_{4380}(721,·)$, $\chi_{4380}(3623,·)$, $\chi_{4380}(1463,·)$, $\chi_{4380}(3001,·)$, $\chi_{4380}(3407,·)$, $\chi_{4380}(2929,·)$, $\chi_{4380}(3647,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | 4.0.7002306000.2$^{2}$, 12.0.1829654030234673819254664000000000.2$^{30}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{3}a^{2}$, $\frac{1}{3}a^{3}$, $\frac{1}{45}a^{4}$, $\frac{1}{45}a^{5}$, $\frac{1}{135}a^{6}$, $\frac{1}{945}a^{7}+\frac{3}{7}a$, $\frac{1}{297675}a^{8}-\frac{2}{945}a^{6}+\frac{2}{315}a^{4}+\frac{8}{49}a^{2}+\frac{1}{3}$, $\frac{1}{297675}a^{9}+\frac{2}{315}a^{5}+\frac{8}{49}a^{3}+\frac{4}{21}a$, $\frac{1}{553165582725}a^{10}-\frac{48301}{61462842525}a^{8}+\frac{1749539}{585360405}a^{6}+\frac{6506102}{1365840945}a^{4}-\frac{26594168}{273168189}a^{2}+\frac{152441}{619429}$, $\frac{1}{3872159079075}a^{11}+\frac{190376}{143413299225}a^{9}-\frac{56201}{117072081}a^{7}-\frac{88885964}{9560886615}a^{5}+\frac{107202496}{1912177323}a^{3}-\frac{466988}{4336003}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $7$ |
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{1372628}$, which has order $21962048$ (assuming GRH)
Unit group
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: |
$\frac{15868}{184388527575}a^{10}+\frac{1114901}{12292568505}a^{8}+\frac{282031}{27874305}a^{6}+\frac{106689938}{273168189}a^{4}+\frac{552189958}{91056063}a^{2}+\frac{31781508}{619429}$, $\frac{128404}{553165582725}a^{10}+\frac{6588748}{26341218225}a^{8}+\frac{19312054}{585360405}a^{6}+\frac{48863305}{30352021}a^{4}+\frac{1269466171}{39024027}a^{2}+\frac{428258731}{1858287}$, $\frac{1607}{6829204725}a^{10}+\frac{45242566}{184388527575}a^{8}+\frac{14656888}{585360405}a^{6}+\frac{247557190}{273168189}a^{4}+\frac{394464894}{30352021}a^{2}+\frac{106538971}{1858287}$, $\frac{4586}{32539151925}a^{10}+\frac{1579639}{10846383975}a^{8}+\frac{468572}{34432965}a^{6}+\frac{31554209}{80343585}a^{4}+\frac{54051350}{16068717}a^{2}+\frac{443239}{109311}$, $\frac{53299}{553165582725}a^{10}+\frac{19677503}{184388527575}a^{8}+\frac{649891}{39024027}a^{6}+\frac{418110481}{455280315}a^{4}+\frac{5258250394}{273168189}a^{2}+\frac{216508148}{1858287}$
| 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: | \( 196804.70719979244 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 196804.70719979244 \cdot 21962048}{2\cdot\sqrt{1829654030234673819254664000000000}}\cr\approx \mathstrut & 3.10865870627553 \end{aligned}\] (assuming GRH)
Galois group
A cyclic group of order 12 |
The 12 conjugacy class representatives for $C_{12}$ |
Character table for $C_{12}$ |
Intermediate fields
\(\Q(\sqrt{365}) \), 3.3.5329.1, 4.0.7002306000.2, 6.6.259133949125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | R | ${\href{/padicField/7.1.0.1}{1} }^{12}$ | ${\href{/padicField/11.12.0.1}{12} }$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.1.0.1}{1} }^{12}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.12.0.1}{12} }$ | ${\href{/padicField/31.12.0.1}{12} }$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.1.0.1}{1} }^{12}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.12.0.1}{12} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(2\)
| 2.12.12.25 | $x^{12} + 12 x^{11} + 60 x^{10} + 160 x^{9} + 308 x^{8} + 736 x^{7} + 2272 x^{6} + 5632 x^{5} + 10608 x^{4} + 15040 x^{3} + 12224 x^{2} + 3584 x + 704$ | $2$ | $6$ | $12$ | $C_{12}$ | $[2]^{6}$ |
\(3\)
| 3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
\(5\)
| 5.12.9.3 | $x^{12} + 75 x^{4} - 375$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
\(73\)
| 73.12.11.8 | $x^{12} + 1533$ | $12$ | $1$ | $11$ | $C_{12}$ | $[\ ]_{12}$ |