Normalized defining polynomial
\( x^{30} - 2x - 2 \)
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)
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Discriminant: | \(113293991873921541159561526868059944925023774709907456\) \(\medspace = 2^{30}\cdot 31\cdot 1493\cdot 294391\cdot 77\!\cdots\!73\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(58.68\) | 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: | \(2\), \(31\), \(1493\), \(294391\), \(77439\!\cdots\!30673\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{10551\!\cdots\!83469}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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)
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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: | $a+1$, $a^{20}+a^{10}+a+1$, $a^{15}-a-1$, $a^{29}+a^{27}+a^{25}+a^{23}+a^{21}+a^{19}+a^{17}-a^{14}-a^{13}-a^{12}-a^{11}-a^{10}-a^{9}-a^{8}-a^{7}-a^{6}-a^{5}-a^{4}-a^{3}-a^{2}-1$, $2a^{29}-a^{28}+a^{26}-2a^{25}+2a^{24}-2a^{23}+3a^{22}-3a^{21}+a^{20}+a^{17}-2a^{16}+2a^{15}-2a^{14}+2a^{13}-2a^{12}+a^{11}+a^{10}-a^{9}-2a^{7}+4a^{6}-2a^{5}-a^{3}+a^{2}-5$, $a^{29}+a^{26}-a^{25}-a^{23}-a^{22}+a^{21}-a^{20}+2a^{19}-2a^{15}-a^{13}+a^{12}+2a^{11}+2a^{9}-a^{8}-a^{5}+a^{4}-a^{3}+a^{2}-3$, $a^{29}-a^{28}+a^{27}-a^{26}+a^{25}-a^{24}+2a^{22}-a^{21}-a^{17}+2a^{16}-a^{15}-a^{14}+a^{11}-a^{10}+2a^{9}-a^{8}-a^{7}+2a^{6}-a^{4}+a^{3}-3a-1$, $a^{29}+a^{26}+a^{24}+a^{22}+a^{20}+a^{18}+a^{16}+2a^{14}+a^{12}+a^{11}+a^{9}+a^{8}+a^{7}+a^{6}+a^{5}+a^{4}+a^{3}+2a^{2}+a-1$, $6a^{29}-6a^{28}+5a^{27}-5a^{26}+5a^{25}-4a^{24}+3a^{23}-3a^{22}+3a^{21}-a^{20}-a^{17}+2a^{16}-2a^{15}+2a^{14}-3a^{13}+3a^{12}-2a^{11}+3a^{10}-3a^{9}+3a^{8}-3a^{7}+2a^{6}-2a^{5}+2a^{4}-2a^{3}+a^{2}+a-11$, $a^{28}-2a^{27}+2a^{26}+2a^{22}-a^{21}+a^{20}+a^{19}+2a^{16}-a^{15}+a^{13}-2a^{11}+a^{10}-2a^{8}-a^{7}+a^{6}-2a^{5}-2a^{4}+a^{3}-3a+1$, $4a^{29}-2a^{28}-a^{27}+3a^{26}-3a^{25}+a^{24}+2a^{23}-4a^{22}+4a^{21}-2a^{20}-a^{19}+3a^{18}-3a^{17}+a^{16}+2a^{15}-4a^{14}+4a^{13}-2a^{12}-a^{11}+3a^{10}-3a^{9}+a^{8}+a^{7}-2a^{6}+a^{5}+2a^{4}-4a^{3}+5a^{2}-3a-9$, $a^{28}-a^{27}+a^{25}-a^{23}+a^{21}-a^{20}-a^{19}+a^{18}+a^{17}-a^{16}+2a^{14}-2a^{12}+a^{10}-2a^{9}-2a^{8}+a^{7}+a^{6}-2a^{5}+2a^{3}+a^{2}-2a+1$, $2a^{29}-2a^{28}+a^{27}-a^{22}+a^{21}-a^{20}+a^{19}-a^{17}+3a^{15}-3a^{14}+2a^{12}-2a^{11}+3a^{9}-3a^{8}-a^{7}+4a^{6}-2a^{5}-2a^{4}+3a^{3}-2a^{2}-a-1$, $2a^{29}+a^{27}-a^{26}-3a^{25}-a^{23}+2a^{22}+a^{21}-2a^{20}-a^{18}+3a^{17}+3a^{16}+a^{15}-4a^{13}+a^{10}-5a^{8}-2a^{7}-a^{6}+3a^{5}+6a^{4}+a^{3}-3a-3$, $a^{29}+2a^{28}-2a^{26}-2a^{24}+4a^{23}-a^{22}+a^{21}-3a^{20}+4a^{17}-a^{16}-4a^{14}-a^{13}+3a^{12}+3a^{11}+2a^{10}-3a^{9}-5a^{8}-2a^{7}+4a^{6}+4a^{5}+5a^{4}-5a^{3}-5a^{2}-4a+1$ (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: | \( 787485177668371.2 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{14}\cdot 787485177668371.2 \cdot 1}{2\cdot\sqrt{113293991873921541159561526868059944925023774709907456}}\cr\approx \mathstrut & 0.699337958041594 \end{aligned}\] (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 | $19{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.13.0.1}{13} }{,}\,{\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $29{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $29{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $25{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | $29{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.11.0.1}{11} }{,}\,{\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ | $30$ | R | $28{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | $17{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | $19{,}\,{\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 $30$ | $30$ | $1$ | $30$ | |||
\(31\) | 31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Deg $28$ | $1$ | $28$ | $0$ | $C_{28}$ | $[\ ]^{28}$ | ||
\(1493\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
\(294391\) | $\Q_{294391}$ | $x$ | $1$ | $1$ | $0$ | 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 $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
\(774\!\cdots\!673\) | 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 $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ |