Normalized defining polynomial
\( x^{30} - 2x^{15} + 2 \)
Invariants
Degree: | $30$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 15]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-3373320308238729216000000000000000000000000000000\) \(\medspace = -\,2^{44}\cdot 3^{30}\cdot 5^{30}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(41.46\) | 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: | $2^{22/15}3^{7/6}5^{23/20}\approx 63.38224664768071$ | ||
Ramified primes: | \(2\), \(3\), \(5\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $14$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( a^{15} - 1 \) (order $4$) | 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: | $a^{5}-1$, $a^{20}-a^{5}-1$, $a^{16}-a-1$, $a-1$, $a^{18}-a^{15}-a^{3}+1$, $a^{18}-a^{3}+1$, $a^{15}-a^{6}-1$, $a^{15}+a^{2}-1$, $a^{21}-a^{18}+a^{15}-a^{6}+a^{3}-1$, $a^{26}-a^{25}+a^{24}+a^{19}-2a^{18}+a^{17}-a^{15}-2a^{11}+2a^{10}-a^{9}-a^{8}+a^{7}-a^{4}+2a^{3}-a+1$, $a^{26}-a^{25}-a^{22}-a^{19}+a^{17}-a^{16}+a^{15}-2a^{11}+a^{9}-a^{8}+a^{7}+a^{4}+2a^{3}-2a^{2}+a-1$, $a^{27}-a^{26}+a^{24}-a^{22}+a^{20}-a^{19}+a^{16}-a^{15}-a^{14}-a^{12}+a^{10}-2a^{9}-2a^{5}+a^{4}-2a+1$, $3a^{29}+2a^{28}+2a^{27}-a^{25}-3a^{24}-2a^{23}-3a^{22}-3a^{21}-3a^{20}-a^{19}-a^{18}+a^{17}+2a^{16}+2a^{15}-2a^{14}-a^{12}+a^{11}+3a^{10}+5a^{9}+4a^{8}+4a^{7}+3a^{6}+a^{5}-2a^{3}-4a^{2}-5a-3$, $a^{29}-4a^{28}+2a^{27}-3a^{25}+5a^{24}-4a^{23}+5a^{22}+a^{21}-a^{20}+4a^{19}-3a^{18}+2a^{17}+a^{16}-5a^{15}+a^{14}+2a^{13}-4a^{12}+2a^{11}-5a^{9}+5a^{8}-10a^{7}+5a^{6}-3a^{5}-2a^{4}+5a^{3}-6a^{2}+5a+3$ (assuming GRH) | 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: | \( 4813633304417.372 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{15}\cdot 4813633304417.372 \cdot 1}{4\cdot\sqrt{3373320308238729216000000000000000000000000000000}}\cr\approx \mathstrut & 0.615293386206380 \end{aligned}\] (assuming GRH)
Galois group
$D_6\times F_5$ (as 30T51):
A solvable group of order 240 |
The 30 conjugacy class representatives for $D_6\times F_5$ |
Character table for $D_6\times F_5$ is not computed |
Intermediate fields
\(\Q(\sqrt{-1}) \), 3.1.108.1, 5.1.50000.1, 6.0.186624.1, 10.0.160000000000.1, 15.1.7174453500000000000000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 30 siblings: | data not computed |
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 | R | R | ${\href{/padicField/7.12.0.1}{12} }^{2}{,}\,{\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.10.0.1}{10} }^{3}$ | ${\href{/padicField/13.12.0.1}{12} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{6}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{5}$ | ${\href{/padicField/23.4.0.1}{4} }^{6}{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{14}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }^{3}$ | ${\href{/padicField/37.12.0.1}{12} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{6}{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.4.0.1}{4} }^{6}{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.4.0.1}{4} }^{6}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{15}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $30$ | $30$ | $1$ | $44$ | |||
\(3\) | 3.6.6.3 | $x^{6} + 18 x^{5} + 120 x^{4} + 386 x^{3} + 723 x^{2} + 732 x + 305$ | $3$ | $2$ | $6$ | $D_{6}$ | $[3/2]_{2}^{2}$ |
3.12.12.23 | $x^{12} + 6 x^{11} + 12 x^{10} + 12 x^{9} + 189 x^{8} + 108 x^{7} + 324 x^{6} + 648 x^{5} + 891 x^{4} + 918 x^{3} + 648 x^{2} + 324 x + 81$ | $3$ | $4$ | $12$ | $S_3 \times C_4$ | $[3/2]_{2}^{4}$ | |
3.12.12.23 | $x^{12} + 6 x^{11} + 12 x^{10} + 12 x^{9} + 189 x^{8} + 108 x^{7} + 324 x^{6} + 648 x^{5} + 891 x^{4} + 918 x^{3} + 648 x^{2} + 324 x + 81$ | $3$ | $4$ | $12$ | $S_3 \times C_4$ | $[3/2]_{2}^{4}$ | |
\(5\) | 5.5.5.2 | $x^{5} + 5 x + 5$ | $5$ | $1$ | $5$ | $F_5$ | $[5/4]_{4}$ |
5.5.5.2 | $x^{5} + 5 x + 5$ | $5$ | $1$ | $5$ | $F_5$ | $[5/4]_{4}$ | |
5.10.10.7 | $x^{10} - 50 x^{7} + 10 x^{6} + 10 x^{5} + 175 x^{4} - 250 x^{3} - 225 x^{2} + 50 x + 25$ | $5$ | $2$ | $10$ | $F_{5}\times C_2$ | $[5/4]_{4}^{2}$ | |
5.10.10.7 | $x^{10} - 50 x^{7} + 10 x^{6} + 10 x^{5} + 175 x^{4} - 250 x^{3} - 225 x^{2} + 50 x + 25$ | $5$ | $2$ | $10$ | $F_{5}\times C_2$ | $[5/4]_{4}^{2}$ |