Normalized defining polynomial
\( x^{30} - x + 2 \)
Invariants
Degree: | $30$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 15]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-110536959857798992796726788865438171785268983873516531\) \(\medspace = -\,16069\cdot 68\!\cdots\!99\) | 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: | $16069^{1/2}6878894757470844034894939875875174048495176045399^{1/2}\approx 3.324709910019203e+26$ | ||
Ramified primes: | \(16069\), \(68788\!\cdots\!45399\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-11053\!\cdots\!16531}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | \( -1 \) (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: | $a^{4}-a^{2}+1$, $a^{20}-a^{10}+1$, $a^{24}-a^{18}+a^{12}-a^{6}+1$, $a^{28}+a^{27}+a^{24}+a^{23}+a^{20}+a^{19}+a^{16}+a^{15}+a^{12}+a^{11}+a^{8}+a^{7}+a^{4}+a^{3}+1$, $a^{28}-a^{22}-a^{21}+a^{19}-a^{16}-a^{13}-a^{12}+a^{10}+a^{9}-a^{7}-a^{3}+a+1$, $2a^{29}+a^{28}-a^{27}+a^{26}-2a^{22}-a^{21}+a^{20}-a^{19}+a^{17}-a^{16}+2a^{15}+2a^{14}-2a^{13}+a^{11}-a^{10}-a^{8}-2a^{7}+3a^{6}-a^{4}+2a^{3}-a^{2}+1$, $a^{29}+2a^{28}-a^{26}-a^{25}+a^{23}-a^{21}+a^{19}+a^{18}-2a^{17}-a^{16}+a^{15}+a^{14}+a^{13}-2a^{12}-2a^{11}+a^{10}+2a^{9}-a^{7}+2a^{5}-a^{3}-a^{2}-a+3$, $a^{29}-2a^{27}+2a^{25}-a^{24}-a^{23}+2a^{22}+2a^{21}-2a^{20}+a^{19}+3a^{18}-2a^{16}+3a^{15}+2a^{14}-2a^{13}+4a^{11}-2a^{9}+3a^{8}+3a^{7}-2a^{6}+4a^{4}+a^{3}-3a^{2}+2a+3$, $a^{29}-a^{28}+a^{26}-a^{25}-a^{22}+a^{20}-2a^{19}+a^{18}+a^{17}-a^{16}+a^{15}-a^{14}-a^{13}+a^{12}-2a^{10}+2a^{9}-a^{8}+2a^{6}-2a^{5}+2a^{3}-a^{2}-a+1$, $a^{29}+a^{27}-2a^{25}+a^{24}-a^{23}-3a^{22}-2a^{20}-a^{19}-2a^{18}-a^{17}+2a^{16}-2a^{15}+a^{14}+4a^{13}+2a^{11}+3a^{10}+a^{9}+2a^{8}-2a^{7}+2a^{6}-5a^{4}+2a^{3}-2a^{2}-3a-1$, $a^{29}-a^{28}-2a^{27}-3a^{26}-3a^{25}-a^{24}-a^{20}-3a^{19}-3a^{18}-2a^{17}-a^{16}+a^{15}+3a^{14}+3a^{13}+3a^{12}+3a^{11}+a^{10}+a^{8}+a^{7}+a^{6}+2a^{5}-3a^{3}-4a^{2}-5a-7$, $a^{29}-2a^{27}-4a^{26}-2a^{25}+3a^{24}+5a^{23}+2a^{22}-a^{21}-a^{20}-a^{19}-a^{18}+a^{14}+2a^{13}+2a^{12}-a^{11}-3a^{10}-3a^{9}-4a^{8}+6a^{6}+4a^{5}-a^{4}-3a^{3}-a^{2}+2a-1$, $3a^{29}+3a^{28}-a^{26}+a^{25}+3a^{24}+2a^{23}+2a^{20}+3a^{19}+a^{18}-2a^{17}-3a^{16}-2a^{15}-2a^{14}-3a^{13}-2a^{12}+a^{11}+2a^{10}-a^{9}-3a^{8}-a^{7}+3a^{6}+3a^{5}-a^{3}+3a^{2}+7a+1$, $a^{28}+a^{27}-a^{25}+a^{21}-2a^{18}-a^{17}+a^{16}-a^{15}+a^{14}+a^{13}-a^{12}-a^{11}+a^{9}+2a^{7}-a^{5}-2a^{3}+2a^{2}+a-1$ (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: | \( 860909290548142.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{15}\cdot 860909290548142.0 \cdot 1}{2\cdot\sqrt{110536959857798992796726788865438171785268983873516531}}\cr\approx \mathstrut & 1.21582667179734 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 265252859812191058636308480000000 |
The 5604 conjugacy class representatives for $S_{30}$ are not computed |
Character table for $S_{30}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | $28{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | $30$ | $22{,}\,{\href{/padicField/5.8.0.1}{8} }$ | $29{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $29{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.7.0.1}{7} }^{2}{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $28{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.5.0.1}{5} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | $25{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $30$ |
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 |
---|---|---|---|---|---|---|---|
\(16069\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
\(687\!\cdots\!399\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ |