Normalized defining polynomial
\( x^{12} - 6x^{11} + 24x^{10} - 60x^{9} + 110x^{8} - 134x^{7} + 114x^{6} - 46x^{5} + 5x^{4} + 36x^{2} - 36x + 19 \)
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: | \(40000000000000000\) \(\medspace = 2^{18}\cdot 5^{16}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(24.18\) | 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: | $2^{3/2}5^{8/5}\approx 37.14471242937835$ | ||
Ramified primes: | \(2\), \(5\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{29}a^{10}+\frac{6}{29}a^{9}+\frac{5}{29}a^{8}+\frac{5}{29}a^{7}+\frac{5}{29}a^{6}-\frac{7}{29}a^{5}+\frac{10}{29}a^{4}-\frac{14}{29}a^{3}-\frac{2}{29}a+\frac{12}{29}$, $\frac{1}{5945}a^{11}-\frac{8}{5945}a^{10}+\frac{541}{1189}a^{9}-\frac{274}{1189}a^{8}-\frac{332}{1189}a^{7}-\frac{1324}{5945}a^{6}-\frac{1748}{5945}a^{5}+\frac{526}{1189}a^{4}-\frac{477}{1189}a^{3}-\frac{563}{1189}a^{2}-\frac{279}{5945}a+\frac{1137}{5945}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: | $\frac{342}{5945}a^{11}-\frac{3146}{5945}a^{10}+\frac{2613}{1189}a^{9}-\frac{7321}{1189}a^{8}+\frac{13269}{1189}a^{7}-\frac{80323}{5945}a^{6}+\frac{53059}{5945}a^{5}-\frac{1656}{1189}a^{4}-\frac{2660}{1189}a^{3}+\frac{72}{1189}a^{2}+\frac{1043}{205}a-\frac{20326}{5945}$, $\frac{674}{5945}a^{11}-\frac{4777}{5945}a^{10}+\frac{3916}{1189}a^{9}-\frac{10467}{1189}a^{8}+\frac{19403}{1189}a^{7}-\frac{122396}{5945}a^{6}+\frac{95718}{5945}a^{5}-\frac{6891}{1189}a^{4}-\frac{3379}{1189}a^{3}-\frac{171}{1189}a^{2}+\frac{30689}{5945}a-\frac{46692}{5945}$, $\frac{2144}{5945}a^{11}-\frac{12027}{5945}a^{10}+\frac{9157}{1189}a^{9}-\frac{21123}{1189}a^{8}+\frac{35253}{1189}a^{7}-\frac{179396}{5945}a^{6}+\frac{116333}{5945}a^{5}-\frac{1068}{1189}a^{4}-\frac{2608}{1189}a^{3}-\frac{3804}{1189}a^{2}+\frac{75249}{5945}a-\frac{45232}{5945}$, $\frac{1991}{5945}a^{11}-\frac{13263}{5945}a^{10}+\frac{10229}{1189}a^{9}-\frac{25654}{1189}a^{8}+\frac{43163}{1189}a^{7}-\frac{238814}{5945}a^{6}+\frac{139417}{5945}a^{5}-\frac{2047}{1189}a^{4}-\frac{11914}{1189}a^{3}-\frac{895}{1189}a^{2}+\frac{63406}{5945}a-\frac{52528}{5945}$, $\frac{6682}{5945}a^{11}-\frac{1299}{205}a^{10}+\frac{28856}{1189}a^{9}-\frac{67253}{1189}a^{8}+\frac{113533}{1189}a^{7}-\frac{593668}{5945}a^{6}+\frac{396619}{5945}a^{5}-\frac{7619}{1189}a^{4}-\frac{9322}{1189}a^{3}-\frac{9482}{1189}a^{2}+\frac{220572}{5945}a-\frac{143776}{5945}$ | 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: | \( 5347.50227078 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 5347.50227078 \cdot 1}{2\cdot\sqrt{40000000000000000}}\cr\approx \mathstrut & 0.822564943320 \end{aligned}\]
Galois group
A non-solvable group of order 60 |
The 5 conjugacy class representatives for $A_5$ |
Character table for $A_5$ |
Intermediate fields
6.2.25000000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 5 sibling: | 5.1.25000000.4 |
Degree 6 sibling: | 6.2.25000000.1 |
Degree 10 sibling: | data not computed |
Degree 15 sibling: | data not computed |
Degree 20 sibling: | data not computed |
Degree 30 sibling: | data not computed |
Minimal sibling: | 5.1.25000000.4 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.5.0.1}{5} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.5.0.1}{5} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.18.59 | $x^{12} + 6 x^{11} + 22 x^{10} + 56 x^{9} + 126 x^{8} + 240 x^{7} + 332 x^{6} - 18 x^{5} - 459 x^{4} - 394 x^{3} - 344 x^{2} + 138 x + 423$ | $4$ | $3$ | $18$ | $A_4$ | $[2, 2]^{3}$ |
\(5\) | 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
5.10.16.12 | $x^{10} + 30 x^{9} + 375 x^{8} + 2560 x^{7} + 10480 x^{6} + 26604 x^{5} + 42380 x^{4} + 45040 x^{3} + 44160 x^{2} + 50640 x + 35972$ | $5$ | $2$ | $16$ | $D_5$ | $[2]^{2}$ |