Normalized defining polynomial
\( x^{31} - x - 2 \)
Invariants
Degree: | $31$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 15]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-18327886165090490685286331351835033630345588173537542144\) \(\medspace = -\,2^{31}\cdot 1433\cdot 3653617274537267\cdot 1630096334391835835360875373\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(60.63\) | 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: | \(2\), \(1433\), \(3653617274537267\), \(1630096334391835835360875373\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-17069\!\cdots\!43806}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$
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)
| |
Fundamental units: | $a^{16}+a+1$, $a^{21}-a^{11}-1$, $a^{26}+a^{21}+a^{16}+a^{11}+a^{6}+a+1$, $a^{28}-a^{25}+a^{22}-a^{19}+a^{16}-a^{13}+a^{10}-a^{7}+a^{4}-a-1$, $a^{30}-a^{29}+a^{28}-a^{27}+a^{26}-a^{25}+a^{24}-a^{23}+a^{22}-a^{21}+a^{20}-a^{19}+a^{18}-a^{17}+a^{16}-a^{15}+a^{14}-a^{13}+a^{12}-a^{11}+a^{10}-a^{9}+a^{8}-a^{7}+a^{6}-a^{5}+a^{4}-a^{3}+a^{2}-a-1$, $a^{25}-a^{19}+a^{13}-a^{7}-1$, $a^{29}-a^{27}+a^{23}-a^{21}+a^{17}-a^{15}+a^{11}-a^{9}+a^{5}-a^{3}-1$, $a^{28}+a^{25}+a^{24}-a^{23}+2a^{21}-2a^{19}+2a^{17}-a^{16}-2a^{15}+a^{13}-a^{12}-a^{11}+a^{4}+a+1$, $2a^{30}+2a^{29}+2a^{28}+a^{27}+a^{26}+a^{25}+a^{23}-2a^{22}-3a^{21}-3a^{20}-3a^{19}-a^{18}+a^{16}+2a^{14}+3a^{13}+3a^{12}+5a^{11}+2a^{10}-2a^{8}-3a^{7}-3a^{6}-3a^{5}-2a^{4}-5a^{3}-3a^{2}-a-1$, $2a^{30}-a^{26}+3a^{25}-2a^{24}+2a^{23}-2a^{22}+2a^{21}-a^{19}+a^{18}+2a^{16}-a^{15}-a^{14}+2a^{13}+3a^{11}-2a^{10}+a^{9}+a^{8}+a^{7}+2a^{6}-2a^{5}+4a^{4}+a^{2}-a-1$, $a^{30}-a^{28}+a^{27}-a^{26}-a^{23}+a^{22}+a^{19}+a^{18}+a^{16}+a^{15}-a^{14}+a^{13}-a^{12}-2a^{9}+a^{8}-a^{6}+a^{5}+a^{4}+a^{3}+a^{2}+a-1$, $a^{29}+a^{26}-a^{24}+a^{22}-a^{21}-a^{20}-a^{18}-a^{17}+a^{16}+2a^{15}-a^{14}+a^{12}-a^{11}+2a^{9}+a^{8}-2a^{7}-a^{6}-2a^{4}+2a^{2}-1$, $2a^{30}-3a^{29}+2a^{28}-2a^{27}+a^{26}-a^{25}-2a^{24}+2a^{23}-a^{22}+2a^{21}-2a^{20}-2a^{17}+a^{16}-a^{15}-2a^{13}-3a^{12}+2a^{11}-2a^{10}+2a^{9}-2a^{8}+a^{7}-a^{6}-2a^{5}+2a^{4}-3a-5$, $a^{30}+a^{29}-2a^{28}+a^{27}-a^{26}+2a^{25}-2a^{24}+a^{23}+a^{21}-a^{19}-a^{17}+2a^{16}-3a^{15}+a^{14}+2a^{12}-2a^{11}-a^{10}+a^{9}+a^{8}+a^{7}-2a^{6}+2a^{5}-a^{4}+2a^{3}-3a^{2}-2a-1$, $5a^{30}-a^{29}-9a^{27}+10a^{26}-4a^{25}-8a^{23}-6a^{22}+14a^{21}+3a^{20}+10a^{19}-10a^{18}-3a^{17}+4a^{16}-a^{15}+4a^{14}-18a^{13}+5a^{11}+14a^{10}+12a^{9}-12a^{8}-6a^{6}+10a^{5}-2a^{4}-17a^{3}-4a^{2}-2a+23$ (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: | \( 6213208850652677.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{15}\cdot 6213208850652677.0 \cdot 1}{2\cdot\sqrt{18327886165090490685286331351835033630345588173537542144}}\cr\approx \mathstrut & 1.36288053289608 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 8222838654177922817725562880000000 |
The 6842 conjugacy class representatives for $S_{31}$ are not computed |
Character table for $S_{31}$ |
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 | $29{,}\,{\href{/padicField/3.2.0.1}{2} }$ | $31$ | ${\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | $23{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.7.0.1}{7} }$ | $20{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.11.0.1}{11} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.7.0.1}{7} }$ | $16{,}\,{\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $31$ | $15{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $27{,}\,{\href{/padicField/41.4.0.1}{4} }$ | $17{,}\,{\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.5.0.1}{5} }$ | $30{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.9.0.1}{9} }^{2}{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.6.0.1}{6} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.2.3.1 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
2.8.8.3 | $x^{8} + 2 x^{7} + 32 x^{6} + 116 x^{5} + 456 x^{4} + 696 x^{3} + 1152 x^{2} + 432 x + 1296$ | $2$ | $4$ | $8$ | $C_2^3: C_4$ | $[2, 2, 2]^{4}$ | |
2.8.8.3 | $x^{8} + 2 x^{7} + 32 x^{6} + 116 x^{5} + 456 x^{4} + 696 x^{3} + 1152 x^{2} + 432 x + 1296$ | $2$ | $4$ | $8$ | $C_2^3: C_4$ | $[2, 2, 2]^{4}$ | |
2.8.8.2 | $x^{8} + 8 x^{7} + 56 x^{6} + 240 x^{5} + 816 x^{4} + 2048 x^{3} + 3776 x^{2} + 4928 x + 3760$ | $2$ | $4$ | $8$ | $C_2^2:C_4$ | $[2, 2]^{4}$ | |
\(1433\) | $\Q_{1433}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{1433}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
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 $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
\(3653617274537267\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $23$ | $1$ | $23$ | $0$ | $C_{23}$ | $[\ ]^{23}$ | ||
\(163\!\cdots\!373\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $29$ | $1$ | $29$ | $0$ | $C_{29}$ | $[\ ]^{29}$ |