Normalized defining polynomial
\( x^{30} - 2 \)
Invariants
Degree: | $30$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 14]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(110536959860366678949888000000000000000000000000000000\) \(\medspace = 2^{59}\cdot 3^{30}\cdot 5^{30}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(58.63\) | 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: | $2^{59/30}3^{7/6}5^{23/20}\approx 89.6360328228267$ | ||
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: | \(\Q(\sqrt{2}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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: | $15$ | 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: | $a+1$, $a-1$, $a^{6}-1$, $a^{3}-1$, $a^{10}-1$, $a^{5}+1$, $a^{15}-1$, $a^{18}+a^{6}+1$, $a^{16}-a^{2}-1$, $a^{24}+a^{21}+a^{18}-a^{9}-a^{6}-2a^{3}-1$, $a^{24}-a^{20}+a^{12}-a^{8}+1$, $a^{26}+a^{24}-a^{22}-a^{20}+a^{16}-a^{14}-a^{12}-a^{10}+a^{4}-1$, $34a^{29}-11a^{28}-44a^{27}-5a^{26}+36a^{25}+3a^{24}-45a^{23}-18a^{22}+41a^{21}+30a^{20}-29a^{19}-28a^{18}+35a^{17}+49a^{16}-19a^{15}-51a^{14}+10a^{13}+50a^{12}-8a^{11}-64a^{10}-17a^{9}+54a^{8}+23a^{7}-53a^{6}-32a^{5}+59a^{4}+59a^{3}-35a^{2}-55a+33$, $18a^{29}-47a^{28}+66a^{27}-16a^{26}-28a^{25}+70a^{24}-59a^{23}+10a^{22}+44a^{21}-84a^{20}+50a^{19}-6a^{18}-66a^{17}+75a^{16}-45a^{15}-24a^{14}+89a^{13}-79a^{12}+48a^{11}+53a^{10}-85a^{9}+88a^{8}-20a^{7}-74a^{6}+92a^{5}-89a^{4}-22a^{3}+85a^{2}-126a+73$, $19a^{29}-24a^{28}-91a^{27}-169a^{26}-235a^{25}-282a^{24}-316a^{23}-338a^{22}-371a^{21}-415a^{20}-429a^{19}-366a^{18}-282a^{17}-237a^{16}-183a^{15}-60a^{14}+83a^{13}+178a^{12}+236a^{11}+296a^{10}+364a^{9}+445a^{8}+523a^{7}+569a^{6}+552a^{5}+528a^{4}+535a^{3}+502a^{2}+345a+153$ (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: | \( 437714372837916.75 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{14}\cdot 437714372837916.75 \cdot 1}{2\cdot\sqrt{110536959860366678949888000000000000000000000000000000}}\cr\approx \mathstrut & 0.393536666125693 \end{aligned}\] (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 $D_6\times F_5$ |
Intermediate fields
\(\Q(\sqrt{2}) \), 3.1.108.1, 5.1.50000.1, 6.2.1492992.4, 10.2.5120000000000.4, 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 | ${\href{/padicField/7.12.0.1}{12} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.10.0.1}{10} }^{3}$ | ${\href{/padicField/13.12.0.1}{12} }^{2}{,}\,{\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.4.0.1}{4} }^{6}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{5}$ | ${\href{/padicField/23.4.0.1}{4} }^{6}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{15}$ | ${\href{/padicField/31.5.0.1}{5} }^{6}$ | ${\href{/padicField/37.12.0.1}{12} }^{2}{,}\,{\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.10.0.1}{10} }^{2}{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{6}{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.4.0.1}{4} }^{6}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{6}{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{15}$ |
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 $30$ | $30$ | $1$ | $59$ | |||
\(3\) | 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 | $x^{12} + 6 x^{11} + 12 x^{10} + 12 x^{9} + 189 x^{8} + 108 x^{7} + 324 x^{6} + 648 x^{5} + 891 x^{4} + 918 x^{3} + 648 x^{2} + 324 x + 81$ | $3$ | $4$ | $12$ | $S_3 \times C_4$ | $[3/2]_{2}^{4}$ | |
\(5\) | 5.10.10.7 | $x^{10} - 50 x^{7} + 10 x^{6} + 10 x^{5} + 175 x^{4} - 250 x^{3} - 225 x^{2} + 50 x + 25$ | $5$ | $2$ | $10$ | $F_{5}\times C_2$ | $[5/4]_{4}^{2}$ |
5.10.10.7 | $x^{10} - 50 x^{7} + 10 x^{6} + 10 x^{5} + 175 x^{4} - 250 x^{3} - 225 x^{2} + 50 x + 25$ | $5$ | $2$ | $10$ | $F_{5}\times C_2$ | $[5/4]_{4}^{2}$ | |
5.10.10.7 | $x^{10} - 50 x^{7} + 10 x^{6} + 10 x^{5} + 175 x^{4} - 250 x^{3} - 225 x^{2} + 50 x + 25$ | $5$ | $2$ | $10$ | $F_{5}\times C_2$ | $[5/4]_{4}^{2}$ |