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: | $[0, 15]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-14130386091738731937078657905788865438171785268983873516531\) \(\medspace = -\,13\cdot 179\cdot 560893979\cdot 1599364314633798641\cdot 6769077172765732520137982927\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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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))
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Galois root discriminant: | $13^{1/2}179^{1/2}560893979^{1/2}1599364314633798641^{1/2}6769077172765732520137982927^{1/2}\approx 1.1887130053860239e+29$ | ||
Ramified primes: | \(13\), \(179\), \(560893979\), \(1599364314633798641\), \(6769077172765732520137982927\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | $\Q(\sqrt{-14130\!\cdots\!16531}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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)
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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)
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Fundamental units: | $a^{24}-a^{23}+2a^{21}-a^{20}-2a^{19}+a^{18}+a^{17}-2a^{16}+a^{14}+a^{13}-a^{12}-a^{11}+3a^{10}-3a^{8}+a^{7}+2a^{6}-3a^{5}+a^{3}-a^{2}+a+1$, $2a^{29}-a^{27}+a^{26}+2a^{25}+a^{24}-a^{23}-2a^{22}+2a^{21}+4a^{20}-3a^{18}-2a^{17}+3a^{16}+3a^{15}-2a^{14}-2a^{13}+2a^{11}+a^{10}-4a^{9}-3a^{8}+2a^{7}+4a^{6}-8a^{4}-4a^{3}+5a^{2}+3a-5$, $3a^{29}+a^{28}+3a^{27}+3a^{26}+2a^{25}+4a^{24}+a^{22}+a^{21}-4a^{20}-a^{19}-4a^{18}-6a^{17}-3a^{16}-7a^{15}-3a^{14}-3a^{13}-5a^{12}+3a^{11}+2a^{9}+7a^{8}+3a^{7}+9a^{6}+6a^{5}+4a^{4}+10a^{3}-a^{2}+3a+1$, $6a^{29}-a^{28}-3a^{27}+5a^{25}-4a^{24}-a^{23}+a^{22}+5a^{21}-7a^{20}+6a^{18}-a^{17}-8a^{16}+7a^{15}+3a^{14}-7a^{13}-a^{12}+8a^{11}-2a^{10}-8a^{9}+6a^{8}+4a^{7}-4a^{6}-8a^{5}+13a^{4}-a^{3}-10a^{2}+13$, $a^{29}+2a^{26}+3a^{25}+3a^{24}+4a^{23}+3a^{22}+3a^{21}+5a^{20}+5a^{19}+3a^{18}+2a^{17}+3a^{16}+3a^{15}+5a^{14}+6a^{13}+3a^{12}+3a^{11}+4a^{10}+4a^{9}+4a^{8}+4a^{7}+a^{6}-2a^{5}+2a^{4}+2a^{3}-a^{2}-a-4$, $a^{29}-3a^{28}-5a^{27}-5a^{26}-2a^{25}+2a^{24}+5a^{23}+6a^{22}+4a^{21}-5a^{19}-6a^{18}-6a^{17}-a^{16}+3a^{15}+7a^{14}+7a^{13}+4a^{12}-2a^{11}-7a^{10}-8a^{9}-7a^{8}+5a^{6}+10a^{5}+8a^{4}+3a^{3}-5a^{2}-9a-11$, $a^{29}+3a^{25}-5a^{24}+6a^{23}-2a^{22}-2a^{21}+8a^{20}-7a^{19}+5a^{18}+a^{17}-4a^{16}+4a^{15}-2a^{14}+a^{13}-2a^{12}+5a^{11}-5a^{10}+7a^{8}-15a^{7}+14a^{6}-7a^{5}-4a^{4}+12a^{3}-12a^{2}+6a-1$, $3a^{29}+a^{27}-5a^{26}+2a^{25}+3a^{24}+a^{23}-6a^{21}+5a^{20}+a^{19}-3a^{17}-5a^{16}+7a^{15}-5a^{12}-a^{11}+9a^{10}-2a^{9}-7a^{7}+3a^{6}+9a^{5}-7a^{4}-9a^{2}+9a+4$, $5a^{29}+6a^{28}+3a^{27}+a^{26}-a^{25}-3a^{24}-5a^{23}-2a^{22}+2a^{20}+7a^{19}+9a^{18}+10a^{17}+9a^{16}+10a^{15}+2a^{14}+a^{13}-3a^{12}-4a^{11}-6a^{10}+a^{9}+a^{8}+5a^{7}+11a^{6}+12a^{5}+9a^{4}+9a^{3}+3a^{2}-6a-13$, $3a^{29}-4a^{28}+6a^{26}-8a^{25}+4a^{24}+2a^{22}-8a^{21}+12a^{20}-9a^{19}+2a^{18}+6a^{16}-13a^{15}+13a^{14}-6a^{13}+a^{12}-6a^{11}+17a^{10}-22a^{9}+16a^{8}-7a^{7}+7a^{6}-18a^{5}+27a^{4}-22a^{3}+6a^{2}+4a-1$, $3a^{29}+a^{28}-3a^{27}-6a^{26}-6a^{25}-4a^{24}-4a^{23}-a^{22}+2a^{21}+6a^{20}+6a^{19}+8a^{18}+5a^{17}-5a^{15}-5a^{14}-7a^{13}-8a^{12}-7a^{11}-a^{10}+a^{9}+7a^{8}+11a^{7}+11a^{6}+4a^{5}-a^{3}-7a^{2}-11a-14$, $a^{29}-a^{28}-a^{27}-2a^{26}-4a^{25}-3a^{24}-4a^{23}-5a^{22}-4a^{21}-5a^{20}-4a^{19}-5a^{18}-6a^{17}-4a^{16}-7a^{15}-7a^{14}-7a^{13}-9a^{12}-7a^{11}-8a^{10}-9a^{9}-5a^{8}-8a^{7}-6a^{6}-6a^{5}-6a^{4}-3a^{3}-5a^{2}-4a-1$, $a^{29}+a^{28}-a^{26}-2a^{25}-a^{24}+a^{22}-a^{20}-a^{19}+2a^{18}+3a^{17}-3a^{15}-a^{14}+2a^{13}+2a^{12}-a^{11}-4a^{10}-3a^{9}+a^{8}+4a^{7}+2a^{6}+3a^{3}+a^{2}-2a-5$, $4a^{29}+10a^{28}-2a^{27}-8a^{26}+a^{25}+14a^{24}+13a^{23}-9a^{22}-14a^{21}+a^{20}+9a^{19}+a^{18}-20a^{17}-12a^{16}+15a^{15}+18a^{14}+a^{13}-14a^{12}-a^{11}+24a^{10}+9a^{9}-20a^{8}-23a^{7}-3a^{6}+23a^{5}-25a^{3}-7a^{2}+22a+31$ (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: | \( 256499963343328640 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{15}\cdot 256499963343328640 \cdot 1}{2\cdot\sqrt{14130386091738731937078657905788865438171785268983873516531}}\cr\approx \mathstrut & 1.01316067079678 \end{aligned}\] (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 | $30$ | $28{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.11.0.1}{11} }{,}\,{\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | R | $27{,}\,{\href{/padicField/17.3.0.1}{3} }$ | $23{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $29{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $30$ | $22{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | $20{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }$ | $19{,}\,{\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(13\) | $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.9.0.1 | $x^{9} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 12 x + 11$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
13.18.0.1 | $x^{18} + 10 x^{11} + 4 x^{10} + 11 x^{9} + 11 x^{8} + 9 x^{7} + 5 x^{6} + 3 x^{5} + 5 x^{4} + 6 x^{3} + 9 x + 2$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | |
\(179\) | 179.2.1.2 | $x^{2} + 179$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
179.3.0.1 | $x^{3} + 4 x + 177$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
Deg $25$ | $1$ | $25$ | $0$ | $C_{25}$ | $[\ ]^{25}$ | ||
\(560893979\) | $\Q_{560893979}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ | ||
\(1599364314633798641\) | $\Q_{1599364314633798641}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ | ||
\(676\!\cdots\!927\) | 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 $21$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ |