Normalized defining polynomial
\( x^{30} - 4x + 4 \)
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: | \(-52511447916628477265270473131940055074976225290092544\) \(\medspace = -\,2^{30}\cdot 23\cdot 21\!\cdots\!97\) | 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: | \(57.19\) | 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: | not computed | ||
Ramified primes: | \(2\), \(23\), \(21263\!\cdots\!44197\) | 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: | $\Q(\sqrt{-48905\!\cdots\!16531}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}$, $\frac{1}{2}a^{16}$, $\frac{1}{2}a^{17}$, $\frac{1}{2}a^{18}$, $\frac{1}{2}a^{19}$, $\frac{1}{2}a^{20}$, $\frac{1}{2}a^{21}$, $\frac{1}{2}a^{22}$, $\frac{1}{2}a^{23}$, $\frac{1}{2}a^{24}$, $\frac{1}{2}a^{25}$, $\frac{1}{2}a^{26}$, $\frac{1}{2}a^{27}$, $\frac{1}{2}a^{28}$, $\frac{1}{2}a^{29}$
Monogenic: | Not computed | |
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: | $\frac{1}{2}a^{15}$, $\frac{1}{2}a^{28}+\frac{1}{2}a^{26}+\frac{1}{2}a^{19}-\frac{1}{2}a^{18}+\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-a^{9}+a^{8}-a^{7}+a^{6}-1$, $\frac{1}{2}a^{23}+\frac{1}{2}a^{22}-\frac{1}{2}a^{20}+\frac{1}{2}a^{18}+\frac{1}{2}a^{17}-\frac{1}{2}a^{15}-a^{9}-a^{8}+a^{6}+a^{5}-a^{4}-a^{3}+a$, $\frac{1}{2}a^{28}+a^{27}+a^{26}+\frac{1}{2}a^{25}+a^{24}+\frac{3}{2}a^{23}+\frac{3}{2}a^{22}+a^{21}+\frac{1}{2}a^{20}+\frac{1}{2}a^{19}+a^{18}+\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-a^{15}-a^{14}-a^{12}-2a^{11}-2a^{10}-a^{9}-a^{7}-2a^{6}-a^{5}+a^{4}+a^{3}-a+1$, $2a^{29}+2a^{28}+2a^{27}+2a^{26}+\frac{3}{2}a^{25}+\frac{3}{2}a^{24}+2a^{23}+\frac{5}{2}a^{22}+2a^{21}+a^{20}+\frac{1}{2}a^{19}+a^{18}+\frac{3}{2}a^{17}+2a^{16}+2a^{15}+a^{14}-a^{12}+a^{11}+2a^{10}+2a^{9}-a^{7}+a^{4}+a^{2}-9$, $\frac{1}{2}a^{28}+\frac{1}{2}a^{27}+a^{25}+a^{24}+a^{23}+a^{22}+\frac{3}{2}a^{21}+a^{20}+a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-a^{14}+a^{11}+a^{9}+2a^{8}-a^{7}+a^{6}-a^{4}-a^{3}-a^{2}-3$, $a^{28}+a^{27}-a^{25}-\frac{1}{2}a^{24}+a^{23}+\frac{1}{2}a^{22}-a^{21}-a^{20}+a^{18}-2a^{16}-\frac{3}{2}a^{15}+a^{14}+2a^{13}-2a^{11}-a^{10}+2a^{9}+2a^{8}-a^{7}-2a^{6}+3a^{4}+2a^{3}-3a^{2}-3a+2$, $\frac{3}{2}a^{29}+\frac{1}{2}a^{28}-\frac{1}{2}a^{27}-\frac{1}{2}a^{26}-a^{23}+\frac{3}{2}a^{21}-\frac{3}{2}a^{19}-\frac{3}{2}a^{18}+\frac{1}{2}a^{17}+a^{16}-a^{15}+2a^{13}-2a^{11}-2a^{10}+2a^{9}+3a^{8}-a^{7}-a^{6}+a^{5}-2a^{3}-3a^{2}+4a-1$, $a^{29}+\frac{1}{2}a^{28}+\frac{1}{2}a^{27}+a^{26}+a^{25}+\frac{1}{2}a^{23}+\frac{3}{2}a^{22}+\frac{1}{2}a^{21}-\frac{1}{2}a^{20}+\frac{1}{2}a^{19}+\frac{3}{2}a^{18}+\frac{1}{2}a^{17}+a^{14}+a^{13}-a^{11}+a^{10}+2a^{9}-a^{8}-a^{7}+a^{6}+2a^{5}-a^{4}-a^{3}+a-3$, $a^{29}+\frac{3}{2}a^{28}+\frac{3}{2}a^{27}+a^{26}+a^{25}+\frac{1}{2}a^{24}+\frac{1}{2}a^{23}+\frac{3}{2}a^{22}+\frac{3}{2}a^{21}+\frac{3}{2}a^{20}+2a^{19}+a^{18}+a^{17}+\frac{3}{2}a^{16}+a^{13}+a^{12}+2a^{11}+2a^{10}+a^{8}+a^{7}-a^{6}+3a^{2}+a-5$, $\frac{1}{2}a^{29}+a^{26}-\frac{1}{2}a^{25}+\frac{3}{2}a^{24}-\frac{1}{2}a^{23}+\frac{1}{2}a^{22}+a^{21}-\frac{1}{2}a^{20}+2a^{19}-a^{18}+\frac{3}{2}a^{17}-a^{15}+2a^{14}-2a^{13}+2a^{12}-a^{11}+a^{9}-3a^{8}+3a^{7}-3a^{6}+2a^{5}-2a^{3}+4a^{2}-6a+3$, $\frac{1}{2}a^{29}+a^{28}+\frac{1}{2}a^{27}-\frac{1}{2}a^{26}-a^{25}+a^{24}+a^{23}-\frac{1}{2}a^{22}-a^{21}-a^{20}+\frac{3}{2}a^{19}+\frac{3}{2}a^{18}-\frac{3}{2}a^{17}-\frac{3}{2}a^{16}-\frac{1}{2}a^{15}+2a^{14}+2a^{13}-2a^{12}-a^{11}+a^{10}+2a^{9}+a^{8}-4a^{7}-2a^{6}+3a^{5}+2a^{4}-4a^{2}-2a+4$, $\frac{1}{2}a^{28}+\frac{1}{2}a^{27}-\frac{1}{2}a^{25}+a^{23}+a^{22}-\frac{1}{2}a^{21}-\frac{3}{2}a^{20}-a^{19}+a^{18}+a^{17}-\frac{1}{2}a^{16}-2a^{15}-a^{14}+a^{13}+2a^{12}-a^{10}-a^{9}+a^{8}+a^{7}-a^{5}-a+1$, $\frac{3}{2}a^{29}-\frac{1}{2}a^{28}+a^{27}+\frac{3}{2}a^{26}-\frac{1}{2}a^{25}+a^{24}-a^{22}-\frac{1}{2}a^{20}+\frac{1}{2}a^{19}-a^{18}-\frac{1}{2}a^{17}+a^{16}-\frac{7}{2}a^{15}+a^{13}-4a^{12}+2a^{11}-4a^{9}+a^{8}-2a^{7}-a^{6}-a^{4}+2a^{3}-5a^{2}-4$ (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)]
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Regulator: | \( 552056500614525.9 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{15}\cdot 552056500614525.9 \cdot 1}{2\cdot\sqrt{52511447916628477265270473131940055074976225290092544}}\cr\approx \mathstrut & 1.13116135947245 \end{aligned}\] (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 $S_{30}$ is not computed |
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 | R | $19{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $29{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $27{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | $30$ | ${\href{/padicField/19.13.0.1}{13} }{,}\,{\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }$ | R | $29{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.11.0.1}{11} }{,}\,{\href{/padicField/41.5.0.1}{5} }$ | ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $29{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $30$ | $30$ | $1$ | $30$ | |||
\(23\) | $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
23.5.0.1 | $x^{5} + 3 x + 18$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
23.6.0.1 | $x^{6} + x^{4} + 9 x^{3} + 9 x^{2} + x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
23.15.0.1 | $x^{15} + 2 x^{6} + 8 x^{5} + 15 x^{4} + 9 x^{3} + 7 x^{2} + 18 x + 18$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | |
\(212\!\cdots\!197\) | $\Q_{21\!\cdots\!97}$ | $x$ | $1$ | $1$ | $0$ | 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 $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ |