Normalized defining polynomial
\( x^{30} - x - 3 \)
Invariants
Degree: | $30$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 14]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(14130386091738737072450964228211134561828214731016126483469\) \(\medspace = 277\cdot 400264301\cdot 5331180702097\cdot 132214342318352251\cdot 180811241978611351\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(86.76\) | 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: | $277^{1/2}400264301^{1/2}5331180702097^{1/2}132214342318352251^{1/2}180811241978611351^{1/2}\approx 1.188713005386024e+29$ | ||
Ramified primes: | \(277\), \(400264301\), \(5331180702097\), \(132214342318352251\), \(180811241978611351\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{14130\!\cdots\!83469}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $a+1$, $a^{15}-2$, $a^{28}+a^{26}+a^{24}+a^{22}+a^{20}+a^{18}+a^{16}+a^{14}+a^{12}+a^{10}+a^{8}+a^{6}+a^{4}+a^{2}+2$, $a^{29}-a^{28}+a^{26}-a^{22}+a^{21}-a^{19}+2a^{18}+a^{17}-2a^{16}+a^{14}-2a^{13}+a^{12}+2a^{11}-a^{10}-a^{8}-a^{7}+a^{6}-a^{4}+3a^{3}-5a+1$, $3a^{29}+a^{28}-2a^{27}-5a^{26}-6a^{25}-6a^{24}-6a^{23}-6a^{22}-3a^{21}+2a^{20}+5a^{19}+7a^{18}+8a^{17}+10a^{16}+10a^{15}+5a^{14}-4a^{12}-7a^{11}-11a^{10}-14a^{9}-13a^{8}-8a^{7}-3a^{6}+a^{5}+9a^{4}+15a^{3}+19a^{2}+17a+10$, $a^{28}-a^{26}+a^{25}+2a^{24}-2a^{23}+a^{22}+3a^{21}-2a^{20}+2a^{18}-a^{17}+a^{15}+a^{14}+a^{12}+3a^{11}-2a^{10}+4a^{8}-2a^{7}-2a^{6}+5a^{5}+a^{4}-2a^{3}+4a^{2}+3a-2$, $a^{28}+a^{27}-a^{26}-a^{25}+2a^{23}-2a^{21}-a^{20}+2a^{19}+3a^{18}-a^{17}-4a^{16}-2a^{15}+4a^{14}+4a^{13}-6a^{11}-3a^{10}+3a^{9}+6a^{8}+a^{7}-5a^{6}-5a^{5}+2a^{4}+6a^{3}+2a^{2}-3a-5$, $3a^{29}+a^{28}-5a^{27}+a^{26}+3a^{25}-4a^{24}-2a^{23}+5a^{22}-a^{21}-5a^{20}+5a^{19}+3a^{18}-6a^{17}+3a^{16}+7a^{15}-5a^{14}-a^{13}+9a^{12}-2a^{11}-5a^{10}+8a^{9}+2a^{8}-8a^{7}+4a^{6}+7a^{5}-10a^{4}-4a^{3}+11a^{2}-7a-14$, $4a^{29}-a^{28}-2a^{27}+5a^{26}-2a^{25}+a^{24}+3a^{23}-5a^{22}+3a^{21}-5a^{19}+4a^{18}-4a^{17}-2a^{16}+4a^{15}-8a^{14}-a^{13}-7a^{11}+6a^{10}-2a^{9}-5a^{8}+7a^{7}-5a^{6}+2a^{5}+7a^{4}-6a^{3}+9a^{2}+6a-7$, $a^{27}-a^{26}-a^{25}+3a^{23}-2a^{22}+a^{20}-3a^{18}+a^{17}+a^{16}+2a^{15}-3a^{14}+a^{13}+2a^{12}-2a^{11}-3a^{10}+4a^{9}-a^{8}-2a^{6}+3a^{5}+a^{4}-4a^{3}-a^{2}+6a-4$, $a^{26}+a^{25}+2a^{24}+3a^{23}+3a^{22}+3a^{21}+3a^{20}+3a^{19}+4a^{18}+3a^{17}+3a^{16}+2a^{15}-2a^{11}-2a^{10}-4a^{9}-6a^{8}-5a^{7}-5a^{6}-6a^{5}-5a^{4}-6a^{3}-6a^{2}-5a-2$, $3a^{29}-2a^{28}+2a^{27}+2a^{25}-a^{24}-a^{23}-a^{22}-5a^{21}-2a^{19}+2a^{18}-5a^{17}+2a^{16}-7a^{15}-a^{14}-4a^{13}+3a^{12}-2a^{11}+4a^{10}+2a^{9}-a^{8}+6a^{5}+2a^{4}+12a^{3}+a^{2}+6a-7$, $5a^{29}-2a^{27}-3a^{26}-3a^{25}+5a^{24}+3a^{23}+2a^{22}-3a^{21}-6a^{20}+3a^{18}+7a^{17}+a^{16}-4a^{15}-6a^{14}-2a^{13}+7a^{12}+6a^{11}+2a^{10}-5a^{9}-11a^{8}+2a^{7}+5a^{6}+10a^{5}+4a^{4}-12a^{3}-6a^{2}-3a+5$, $14a^{28}+3a^{27}-12a^{26}-3a^{25}-a^{24}+16a^{23}-8a^{22}-6a^{21}-7a^{20}+9a^{19}+19a^{18}-9a^{17}-12a^{16}-20a^{15}+20a^{14}+15a^{13}+2a^{12}-14a^{11}-18a^{10}+18a^{9}+7a^{8}+6a^{7}-30a^{6}-2a^{5}+26a^{4}+19a^{3}-2a^{2}-46a-4$, $5a^{29}+a^{28}+2a^{27}-5a^{26}-6a^{25}-7a^{24}-9a^{23}-8a^{22}-5a^{21}-5a^{20}+5a^{19}+4a^{18}+12a^{17}+15a^{16}+11a^{15}+10a^{14}+7a^{13}-5a^{12}-6a^{11}-11a^{10}-19a^{9}-13a^{8}-19a^{7}-9a^{6}-5a^{5}+4a^{4}+16a^{3}+24a^{2}+22a+26$ (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: | \( 275535081946504500 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{14}\cdot 275535081946504500 \cdot 1}{2\cdot\sqrt{14130386091738737072450964228211134561828214731016126483469}}\cr\approx \mathstrut & 0.692864070328479 \end{aligned}\] (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 | $30$ | $28{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | $22{,}\,{\href{/padicField/5.8.0.1}{8} }$ | $27{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $30$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | $29{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $28{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.11.0.1}{11} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $26{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.8.0.1}{8} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }$ | $30$ |
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 |
---|---|---|---|---|---|---|---|
\(277\) | $\Q_{277}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $25$ | $1$ | $25$ | $0$ | $C_{25}$ | $[\ ]^{25}$ | ||
\(400264301\) | $\Q_{400264301}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{400264301}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
\(5331180702097\) | $\Q_{5331180702097}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{5331180702097}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | ||
\(132214342318352251\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $20$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ | ||
\(180811241978611351\) | $\Q_{180811241978611351}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{180811241978611351}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ |