Normalized defining polynomial
\( x^{30} - 3x - 3 \)
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: | \(14659049900676850570461305674795650464488012557052441857381\) \(\medspace = 3^{30}\cdot 883\cdot 468509\cdot 30594564677969\cdot 5625294473757709912283\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(86.87\) | 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: | not computed | ||
Ramified primes: | \(3\), \(883\), \(468509\), \(30594564677969\), \(5625294473757709912283\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{71198\!\cdots\!83469}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $15$ | 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: | $a+1$, $a^{15}-a-2$, $a^{29}-2a^{28}-2a^{27}+3a^{26}+2a^{25}-2a^{24}-3a^{23}+2a^{22}+3a^{21}-a^{20}-3a^{19}+a^{18}+4a^{17}-4a^{15}-a^{14}+5a^{13}+3a^{12}-4a^{11}-3a^{10}+5a^{9}+5a^{8}-5a^{7}-6a^{6}+6a^{5}+9a^{4}-4a^{3}-10a^{2}+4a+8$, $2a^{29}-a^{28}+a^{27}-2a^{26}+2a^{25}-a^{24}+a^{23}-3a^{22}+3a^{21}-a^{19}-a^{18}+2a^{17}-a^{15}-2a^{14}+3a^{13}-2a^{11}+2a^{9}-a^{8}-a^{6}+2a^{5}-a^{4}+a^{2}-a-7$, $a^{29}-a^{28}+a^{27}-a^{26}+a^{25}-a^{22}+a^{21}+a^{20}-a^{19}-a^{18}+a^{16}-a^{15}-a^{11}+2a^{10}+a^{9}-a^{8}-a^{7}+2a^{6}+a^{5}-2a^{4}-a^{3}+a^{2}-5$, $3a^{29}-4a^{28}-5a^{27}-2a^{26}+4a^{25}+7a^{24}+a^{23}-4a^{22}-9a^{21}-a^{20}+5a^{19}+11a^{18}+a^{17}-7a^{16}-12a^{15}-a^{14}+9a^{13}+13a^{12}+a^{11}-11a^{10}-14a^{9}-a^{8}+13a^{7}+16a^{6}+a^{5}-17a^{4}-18a^{3}+a^{2}+22a+10$, $a^{29}-4a^{28}+3a^{27}-a^{26}-3a^{25}+a^{24}+a^{23}-4a^{22}+a^{21}-a^{20}-2a^{18}-2a^{17}+a^{16}-5a^{14}+a^{12}-2a^{11}-5a^{10}+a^{9}+a^{8}-6a^{7}-2a^{6}+2a^{5}-5a^{4}-2a^{3}-3a^{2}-2a-2$, $a^{29}+2a^{27}+a^{26}-2a^{25}-a^{23}-3a^{22}-2a^{21}-a^{20}-a^{19}-a^{18}+2a^{17}+3a^{16}+a^{15}+4a^{14}+5a^{13}-a^{12}+2a^{11}+a^{10}-5a^{9}-4a^{8}-3a^{7}-6a^{6}-5a^{5}-a^{4}+2a^{3}+6a+8$, $a^{28}-a^{27}+2a^{26}+a^{25}+a^{24}+3a^{22}+a^{21}+a^{19}+2a^{18}-a^{16}+a^{15}-2a^{13}-3a^{12}-2a^{10}-4a^{9}-3a^{8}-4a^{6}-5a^{5}-a^{4}+a^{3}-3a^{2}-a+4$, $2a^{29}-3a^{28}-a^{27}+2a^{26}+2a^{25}-a^{24}-3a^{23}+a^{22}+3a^{21}-a^{20}-4a^{19}+5a^{17}-4a^{15}+3a^{13}+3a^{12}-6a^{11}-4a^{10}+6a^{9}+3a^{8}-4a^{7}-5a^{6}+6a^{5}+6a^{4}-4a^{3}-8a^{2}+5$, $2a^{29}+a^{28}-2a^{27}-2a^{26}+4a^{25}-2a^{23}-a^{22}+a^{21}+3a^{20}-2a^{19}-4a^{18}+2a^{17}+4a^{16}-a^{15}-5a^{14}+2a^{13}+4a^{12}-a^{11}-3a^{10}-3a^{9}+7a^{8}+4a^{7}-9a^{6}-2a^{5}+6a^{4}+2a^{3}-5a^{2}-6a-1$, $a^{29}-4a^{28}+a^{27}+a^{26}-3a^{25}-2a^{24}+4a^{23}-3a^{22}-4a^{21}+3a^{20}-6a^{18}+a^{17}+a^{16}-6a^{15}-a^{14}+3a^{13}-7a^{12}-3a^{11}+5a^{10}-4a^{9}-8a^{8}+5a^{7}-10a^{5}+3a^{3}-10a^{2}-5a+1$, $3a^{29}-9a^{28}+11a^{27}-3a^{26}-8a^{25}+11a^{24}-7a^{23}+3a^{22}+4a^{21}-14a^{20}+14a^{19}-a^{18}-11a^{17}+12a^{16}-9a^{15}+5a^{14}+6a^{13}-19a^{12}+16a^{11}+3a^{10}-13a^{9}+11a^{8}-12a^{7}+8a^{6}+13a^{5}-27a^{4}+12a^{3}+8a^{2}-13a+5$, $2a^{28}+a^{27}+3a^{26}+2a^{24}+4a^{22}+2a^{20}-a^{19}+a^{18}-a^{17}-3a^{15}-2a^{14}-2a^{13}-3a^{12}-3a^{11}-5a^{10}-5a^{9}-5a^{8}-a^{7}-3a^{6}-a^{5}-4a^{4}+2a^{2}+5a+1$, $a^{29}-a^{28}-a^{26}-a^{23}-a^{20}-3a^{18}-2a^{17}+a^{16}+a^{15}+a^{14}-a^{13}-2a^{12}+a^{11}+a^{9}+3a^{8}+a^{7}+3a^{6}+3a^{5}-a^{3}-a^{2}+4a+5$ (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: | \( 210282023224939900 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{14}\cdot 210282023224939900 \cdot 1}{2\cdot\sqrt{14659049900676850570461305674795650464488012557052441857381}}\cr\approx \mathstrut & 0.519155414702678 \end{aligned}\] (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 | $30$ | R | ${\href{/padicField/5.11.0.1}{11} }{,}\,{\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | $27{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $30$ | $22{,}\,{\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $26{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.10.0.1}{10} }^{3}$ | ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.11.0.1}{11} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $25{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/41.11.0.1}{11} }$ | $17{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/47.13.0.1}{13} }$ | $22{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | Deg $30$ | $30$ | $1$ | $30$ | |||
\(883\) | $\Q_{883}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $27$ | $1$ | $27$ | $0$ | $C_{27}$ | $[\ ]^{27}$ | ||
\(468509\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | ||
\(30594564677969\) | $\Q_{30594564677969}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
\(562\!\cdots\!283\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |