Normalized defining polynomial
\( x^{30} - 3x - 2 \)
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)
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Discriminant: | \(528774345897976432375444495795650464488012557052441857381\) \(\medspace = 3^{30}\cdot 31\cdot 1769381441963\cdot 46821958249121311829160660473\) | 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: | \(77.76\) | 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: | not computed | ||
Ramified primes: | \(3\), \(31\), \(1769381441963\), \(46821958249121311829160660473\) | 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: | $\Q(\sqrt{25682\!\cdots\!83469}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$
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: | $a^{29}+a^{27}+a^{25}+a^{23}+a^{21}+a^{19}+a^{18}+a^{17}+a^{16}+a^{15}+a^{14}+a^{13}+a^{12}+a^{11}+a^{10}+a^{9}+2a^{8}+a^{7}+2a^{6}+a^{5}+2a^{4}+a^{3}+2a^{2}+a-1$, $a^{29}+a^{28}+a^{27}+a^{26}+a^{25}+a^{24}+a^{23}+a^{22}+a^{21}+a^{20}+a^{19}+a^{18}+a^{17}+a^{16}+a^{15}+a^{14}+a^{13}+a^{12}+a^{11}+a^{10}+a^{9}+a^{8}+a^{7}+a^{6}+a^{5}+a^{4}+a^{3}+a^{2}+2a+1$, $3a^{29}-2a^{28}-a^{25}-2a^{24}-2a^{21}+a^{20}+2a^{19}-a^{18}+2a^{17}+3a^{16}+2a^{13}-a^{12}-4a^{11}-2a^{9}-5a^{8}+a^{6}-a^{5}+2a^{4}+6a^{3}+2a^{2}+a-3$, $2a^{29}-2a^{28}+2a^{27}-a^{26}+a^{24}-a^{23}+a^{22}-a^{20}+2a^{19}-3a^{18}+3a^{17}-2a^{16}+a^{15}+a^{14}-2a^{13}+3a^{12}-3a^{11}+3a^{10}-a^{9}+a^{8}+2a^{7}-a^{6}+2a^{5}+3a^{2}-2a-1$, $a^{27}-a^{26}-a^{25}+a^{24}+2a^{23}-2a^{22}-2a^{21}+2a^{20}+2a^{19}-2a^{18}-a^{17}+a^{16}+a^{13}-a^{12}-2a^{11}+2a^{10}+2a^{9}-2a^{8}-2a^{7}+2a^{6}+a^{5}-a^{4}-a^{3}+a^{2}-1$, $a^{29}+a^{26}-a^{24}-a^{20}+a^{18}-a^{17}+2a^{15}+a^{14}+a^{12}+a^{11}-2a^{9}+a^{7}-2a^{6}-a^{5}+a^{4}-a^{2}+a+1$, $3a^{29}+4a^{28}+3a^{27}-a^{25}-a^{24}+a^{23}+5a^{22}+6a^{21}+4a^{20}+a^{19}-3a^{18}-4a^{17}+a^{16}+6a^{15}+10a^{14}+11a^{13}+4a^{12}-3a^{11}-6a^{10}-5a^{9}+3a^{8}+13a^{7}+15a^{6}+12a^{5}+4a^{4}-6a^{3}-6a^{2}+a+1$, $2a^{29}+a^{28}-2a^{26}-2a^{25}+2a^{23}+2a^{22}+a^{21}-a^{20}-3a^{19}-3a^{18}-a^{17}+3a^{16}+6a^{15}+5a^{14}-5a^{12}-7a^{11}-5a^{10}+6a^{8}+9a^{7}+6a^{6}-2a^{5}-8a^{4}-9a^{3}-5a^{2}+2a+1$, $a^{29}-a^{28}+2a^{26}-a^{25}-2a^{24}+2a^{22}-a^{21}-2a^{20}+3a^{19}+2a^{18}-2a^{17}-2a^{16}+2a^{15}+a^{14}-4a^{13}-a^{12}+3a^{11}+a^{10}-a^{9}+a^{8}+3a^{7}-2a^{6}-3a^{5}+a^{4}-a^{3}-2a^{2}+a+1$, $2a^{29}-2a^{28}+2a^{27}-a^{26}+2a^{25}-3a^{24}+a^{23}+2a^{21}+a^{19}-2a^{18}-a^{17}+a^{16}+a^{15}-a^{13}-a^{12}-3a^{11}+3a^{10}+2a^{9}+2a^{8}-3a^{7}+2a^{6}-3a^{5}+3a^{4}+2a^{3}+3a^{2}-7a-7$, $a^{29}-2a^{27}+2a^{26}+a^{24}-2a^{22}+2a^{21}-a^{20}+a^{19}-a^{18}-3a^{17}+2a^{16}-a^{15}+3a^{14}-2a^{13}-2a^{12}+3a^{11}+4a^{9}-4a^{8}-a^{7}+2a^{6}+2a^{4}-7a^{3}-a^{2}+2a-1$, $19a^{29}-12a^{28}+8a^{27}-5a^{26}+2a^{25}-a^{24}+a^{22}-a^{21}+a^{20}+a^{17}-2a^{16}+2a^{15}-2a^{14}+3a^{13}-2a^{12}+a^{11}-a^{10}+2a^{8}-2a^{7}+2a^{6}-3a^{5}+2a^{4}-2a^{3}-a-57$, $56a^{29}-37a^{28}+26a^{27}-17a^{26}+11a^{25}-7a^{24}+4a^{23}-3a^{22}+a^{21}-2a^{20}+2a^{19}-2a^{18}+a^{17}+a^{15}+2a^{14}+a^{11}-a^{10}-a^{9}-a^{8}-2a^{7}-3a^{5}-2a^{4}+a^{3}+2a-167$, $3a^{29}+a^{28}-2a^{27}-3a^{26}-a^{25}+2a^{24}+4a^{23}-3a^{21}-4a^{20}-a^{19}+4a^{18}+4a^{17}-5a^{15}-5a^{14}+6a^{12}+6a^{11}-a^{10}-7a^{9}-6a^{8}+a^{7}+9a^{6}+8a^{5}-2a^{4}-9a^{3}-8a^{2}+3a+3$, $7a^{29}-7a^{28}+7a^{27}-6a^{26}+5a^{25}-4a^{24}+a^{23}+a^{22}-4a^{21}+7a^{20}-8a^{19}+9a^{18}-9a^{17}+6a^{16}-5a^{15}+2a^{14}+a^{13}-3a^{12}+7a^{11}-10a^{10}+11a^{9}-13a^{8}+9a^{7}-6a^{6}+2a^{5}+5a^{4}-8a^{3}+9a^{2}-12a-13$ (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: | \( 96404046034030270 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{14}\cdot 96404046034030270 \cdot 1}{2\cdot\sqrt{528774345897976432375444495795650464488012557052441857381}}\cr\approx \mathstrut & 1.25316491030658 \end{aligned}\] (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 | $28{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | R | $22{,}\,{\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.13.0.1}{13} }{,}\,{\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $29{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $29{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/23.12.0.1}{12} }$ | $30$ | R | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.5.0.1}{5} }$ | $28{,}\,{\href{/padicField/41.2.0.1}{2} }$ | $26{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $30$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.6.6.1 | $x^{6} - 6 x^{5} + 24 x^{4} + 6 x^{3} + 18 x + 9$ | $3$ | $2$ | $6$ | $C_3^2:C_4$ | $[3/2, 3/2]_{2}^{2}$ |
3.12.12.18 | $x^{12} - 36 x^{11} + 438 x^{10} + 120 x^{9} - 4563 x^{8} + 1188 x^{7} + 22410 x^{6} + 10692 x^{5} - 17658 x^{4} - 3780 x^{3} + 6804 x^{2} - 1296 x + 81$ | $3$ | $4$ | $12$ | 12T46 | $[3/2, 3/2]_{2}^{4}$ | |
3.12.12.10 | $x^{12} - 24 x^{11} + 306 x^{10} - 2004 x^{9} + 7236 x^{8} - 4374 x^{7} - 1458 x^{6} + 5832 x^{5} - 1836 x^{3} + 324 x^{2} + 486 x + 81$ | $3$ | $4$ | $12$ | 12T173 | $[3/2, 3/2, 3/2, 3/2]_{2}^{4}$ | |
\(31\) | 31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Deg $28$ | $1$ | $28$ | $0$ | $C_{28}$ | $[\ ]^{28}$ | ||
\(1769381441963\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
\(468\!\cdots\!473\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $20$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ |