Normalized defining polynomial
\( x^{30} - 4x - 1 \)
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)
| |
Discriminant: | \(2757032013554862401424586100944031593515138235298081\) \(\medspace = 1657\cdot 56091358577718162989\cdot 29663564839667553243668426797\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(51.84\) | 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: | $1657^{1/2}56091358577718162989^{1/2}29663564839667553243668426797^{1/2}\approx 5.250744721994074e+25$ | ||
Ramified primes: | \(1657\), \(56091358577718162989\), \(29663564839667553243668426797\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{27570\!\cdots\!98081}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{7}$, $\frac{1}{2}a^{23}-\frac{1}{2}a^{8}$, $\frac{1}{2}a^{24}-\frac{1}{2}a^{9}$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{10}$, $\frac{1}{2}a^{26}-\frac{1}{2}a^{11}$, $\frac{1}{2}a^{27}-\frac{1}{2}a^{12}$, $\frac{1}{2}a^{28}-\frac{1}{2}a^{13}$, $\frac{1}{2}a^{29}-\frac{1}{2}a^{14}$
Monogenic: | Not computed | |
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: | $a$, $\frac{1}{2}a^{15}-\frac{1}{2}$, $\frac{1}{2}a^{15}-a^{10}+a^{5}-\frac{1}{2}$, $\frac{1}{2}a^{15}-a^{12}+a^{9}-a^{6}+a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{29}+\frac{1}{2}a^{14}+a^{13}+a^{12}+a^{11}+a^{10}+a^{9}+a^{8}+a^{7}+a^{6}+a^{5}+a^{4}+a^{3}+a^{2}+a-1$, $\frac{1}{2}a^{29}+\frac{1}{2}a^{28}+\frac{1}{2}a^{27}+\frac{1}{2}a^{26}-\frac{1}{2}a^{23}-\frac{1}{2}a^{22}-\frac{1}{2}a^{21}-\frac{1}{2}a^{20}+\frac{1}{2}a^{17}+\frac{1}{2}a^{16}+\frac{1}{2}a^{15}+\frac{1}{2}a^{14}+\frac{1}{2}a^{13}+\frac{1}{2}a^{12}+\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{3}{2}a^{6}-\frac{3}{2}a^{5}-2a^{4}-a^{3}-\frac{1}{2}a^{2}+\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{29}+\frac{1}{2}a^{18}-\frac{1}{2}a^{15}+\frac{1}{2}a^{14}-a^{11}-a^{7}-a^{6}-\frac{1}{2}a^{3}-a^{2}-\frac{3}{2}$, $\frac{3}{2}a^{29}-\frac{1}{2}a^{28}+\frac{1}{2}a^{26}+\frac{1}{2}a^{25}-a^{24}+a^{22}-\frac{3}{2}a^{21}-\frac{1}{2}a^{20}-\frac{1}{2}a^{18}-a^{17}-a^{16}+\frac{1}{2}a^{15}-\frac{3}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}+\frac{1}{2}a^{10}-a^{9}+a^{8}+a^{7}+\frac{1}{2}a^{6}+\frac{1}{2}a^{5}+a^{4}+\frac{7}{2}a^{3}-a^{2}+a-\frac{3}{2}$, $\frac{3}{2}a^{29}-a^{28}+\frac{1}{2}a^{27}-\frac{1}{2}a^{26}+\frac{1}{2}a^{25}-\frac{1}{2}a^{22}+\frac{1}{2}a^{21}-\frac{1}{2}a^{20}+\frac{1}{2}a^{19}-a^{18}+\frac{1}{2}a^{17}+\frac{1}{2}a^{16}-\frac{1}{2}a^{15}+\frac{1}{2}a^{14}-a^{13}+\frac{3}{2}a^{12}-\frac{3}{2}a^{11}+\frac{1}{2}a^{10}-a^{9}+a^{8}-\frac{1}{2}a^{7}+\frac{1}{2}a^{6}-\frac{3}{2}a^{5}+\frac{5}{2}a^{4}-2a^{3}+\frac{3}{2}a^{2}-\frac{7}{2}a-\frac{5}{2}$, $\frac{1}{2}a^{27}+a^{25}+a^{23}+\frac{1}{2}a^{22}+\frac{1}{2}a^{20}-\frac{1}{2}a^{19}+\frac{1}{2}a^{15}+a^{13}+\frac{1}{2}a^{12}+a^{11}+a^{10}-\frac{3}{2}a^{7}-a^{6}-\frac{5}{2}a^{5}-\frac{5}{2}a^{4}-2a^{3}-4a^{2}-a-\frac{3}{2}$, $\frac{1}{2}a^{28}-\frac{1}{2}a^{26}+\frac{1}{2}a^{24}-\frac{1}{2}a^{19}+\frac{1}{2}a^{17}-a^{16}-a^{15}+\frac{3}{2}a^{13}-a^{12}-\frac{1}{2}a^{11}+\frac{3}{2}a^{9}+\frac{1}{2}a^{4}+a^{3}+\frac{1}{2}a^{2}-2a$, $\frac{1}{2}a^{29}-\frac{1}{2}a^{28}-\frac{1}{2}a^{27}+\frac{1}{2}a^{26}+\frac{1}{2}a^{20}+\frac{1}{2}a^{19}-\frac{3}{2}a^{18}-\frac{1}{2}a^{17}+a^{16}+\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}+\frac{1}{2}a^{12}+\frac{1}{2}a^{11}+a^{10}-a^{9}-2a^{8}+a^{7}+a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}+\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{3}{2}a^{29}-\frac{1}{2}a^{27}-\frac{1}{2}a^{26}+\frac{1}{2}a^{24}-\frac{1}{2}a^{22}+\frac{1}{2}a^{20}+\frac{1}{2}a^{19}+\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-a^{16}-\frac{1}{2}a^{15}+\frac{1}{2}a^{14}+a^{13}-\frac{1}{2}a^{12}-\frac{3}{2}a^{11}+a^{10}+\frac{5}{2}a^{9}-\frac{5}{2}a^{7}-2a^{6}+\frac{1}{2}a^{5}+\frac{5}{2}a^{4}-\frac{1}{2}a^{3}-\frac{7}{2}a^{2}+a-\frac{3}{2}$, $a^{29}-a^{27}+\frac{3}{2}a^{26}-\frac{3}{2}a^{24}+\frac{3}{2}a^{23}-a^{21}+2a^{20}-\frac{1}{2}a^{19}-a^{18}+2a^{17}-a^{16}-\frac{1}{2}a^{15}+2a^{14}-2a^{13}+\frac{5}{2}a^{11}-3a^{10}-\frac{1}{2}a^{9}+\frac{5}{2}a^{8}-3a^{7}+2a^{5}-\frac{7}{2}a^{4}+a^{3}+2a^{2}-4a-\frac{5}{2}$, $\frac{3}{2}a^{29}+\frac{1}{2}a^{28}-\frac{1}{2}a^{26}-a^{25}-a^{24}-a^{23}-\frac{1}{2}a^{22}+\frac{1}{2}a^{21}+a^{20}+\frac{3}{2}a^{19}+\frac{3}{2}a^{18}-\frac{1}{2}a^{16}-\frac{3}{2}a^{15}-\frac{3}{2}a^{14}-\frac{1}{2}a^{13}+\frac{3}{2}a^{11}+3a^{10}+2a^{9}+2a^{8}+\frac{1}{2}a^{7}-\frac{3}{2}a^{6}-2a^{5}-\frac{5}{2}a^{4}-\frac{3}{2}a^{3}+a^{2}+\frac{5}{2}a-\frac{3}{2}$ (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: | \( 72802108759548.89 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{14}\cdot 72802108759548.89 \cdot 1}{2\cdot\sqrt{2757032013554862401424586100944031593515138235298081}}\cr\approx \mathstrut & 0.414449106124039 \end{aligned}\] (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 | ${\href{/padicField/2.8.0.1}{8} }^{2}{,}\,{\href{/padicField/2.4.0.1}{4} }^{3}{,}\,{\href{/padicField/2.2.0.1}{2} }$ | $30$ | $29{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.13.0.1}{13} }{,}\,{\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.14.0.1}{14} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $28{,}\,{\href{/padicField/31.2.0.1}{2} }$ | $24{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $19{,}\,{\href{/padicField/47.11.0.1}{11} }$ | $17{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | $16{,}\,{\href{/padicField/59.13.0.1}{13} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(1657\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $28$ | $1$ | $28$ | $0$ | $C_{28}$ | $[\ ]^{28}$ | ||
\(56091358577718162989\) | $\Q_{56091358577718162989}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $24$ | $1$ | $24$ | $0$ | $C_{24}$ | $[\ ]^{24}$ | ||
\(296\!\cdots\!797\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $28$ | $1$ | $28$ | $0$ | $C_{28}$ | $[\ ]^{28}$ |