Normalized defining polynomial
\( x^{31} - x - 3 \)
Invariants
Degree: | $31$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 15]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-3514391585695303086629141906098469962323789296982843961159719\) \(\medspace = -\,3^{31}\cdot 571\cdot 99\!\cdots\!87\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(89.76\) | 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))
| |
Galois root discriminant: | not computed | ||
Ramified primes: | \(3\), \(571\), \(99644\!\cdots\!55887\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-17069\!\cdots\!34431}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $a^{16}-a-1$, $a^{21}-a^{11}-1$, $a^{26}+a^{21}-a^{11}-a^{6}+1$, $a^{30}-a^{28}+a^{24}-a^{22}+a^{18}-a^{16}+a^{15}+a^{14}-2a^{13}-a^{12}+a^{11}+a^{9}+a^{8}-2a^{7}-a^{6}+a^{5}+a^{3}+a^{2}-2a-1$, $a^{30}-a^{28}+a^{27}-a^{25}+a^{24}-a^{22}+a^{21}-a^{19}+a^{18}-a^{16}+2a^{14}-2a^{13}+2a^{11}-2a^{10}+2a^{8}-2a^{7}+2a^{5}-2a^{4}+2a^{2}-2a-2$, $2a^{28}+a^{27}-3a^{26}-a^{25}+2a^{24}+a^{23}-3a^{22}-a^{21}+4a^{20}+2a^{19}-2a^{18}-3a^{17}+3a^{13}+2a^{12}-2a^{11}-4a^{10}+a^{9}+4a^{8}-3a^{7}-4a^{6}+3a^{5}+8a^{4}-a^{3}-7a^{2}-2a+1$, $8a^{30}-5a^{28}-9a^{27}-16a^{26}-15a^{25}-16a^{24}-13a^{23}-5a^{22}+9a^{20}+17a^{19}+20a^{18}+24a^{17}+21a^{16}+14a^{15}+7a^{14}-7a^{13}-16a^{12}-26a^{11}-33a^{10}-30a^{9}-29a^{8}-16a^{7}-3a^{6}+11a^{5}+31a^{4}+37a^{3}+48a^{2}+46a+25$, $8a^{29}+7a^{28}+a^{27}-10a^{25}-7a^{24}-a^{23}+a^{22}+11a^{21}+5a^{20}-5a^{18}-15a^{17}-5a^{16}+8a^{14}+18a^{13}+6a^{12}+a^{11}-11a^{10}-18a^{9}-4a^{8}+16a^{6}+19a^{5}+2a^{4}-3a^{3}-22a^{2}-20a-1$, $2a^{30}-5a^{29}+3a^{28}-3a^{27}+3a^{25}+2a^{24}-2a^{23}+2a^{21}-8a^{20}+4a^{19}-a^{18}+2a^{17}+a^{16}+4a^{15}-7a^{14}-2a^{13}+3a^{12}-7a^{11}+5a^{10}+3a^{9}+3a^{8}-4a^{7}+5a^{6}-11a^{5}-a^{4}+7a^{3}-a^{2}+3a+4$, $2a^{30}+3a^{29}+2a^{27}+2a^{26}-a^{25}-2a^{22}-a^{21}-a^{19}-a^{18}-a^{16}-3a^{15}-2a^{14}-2a^{13}-5a^{12}-2a^{11}-4a^{9}+a^{8}+3a^{7}-3a^{6}+3a^{5}+4a^{4}-3a^{3}+5a^{2}+6a-2$, $5a^{30}-2a^{29}+4a^{28}-5a^{27}-a^{26}-5a^{25}-9a^{24}-5a^{23}-16a^{22}-7a^{21}-16a^{20}-10a^{19}-10a^{18}-14a^{17}-4a^{16}-14a^{15}-5a^{13}+2a^{12}+9a^{11}+5a^{10}+20a^{9}+11a^{8}+22a^{7}+20a^{6}+20a^{5}+30a^{4}+19a^{3}+30a^{2}+16a+11$, $2a^{30}+5a^{29}+2a^{28}+4a^{27}+2a^{26}+2a^{25}+8a^{24}+2a^{23}+a^{22}+6a^{21}+6a^{20}+5a^{19}+7a^{17}+9a^{16}+a^{15}+7a^{14}+7a^{13}+5a^{12}+8a^{11}+6a^{10}+9a^{9}+5a^{8}+7a^{7}+16a^{6}+3a^{5}+7a^{4}+14a^{3}+9a^{2}+12a+5$, $2a^{30}-3a^{29}-a^{28}+2a^{27}+4a^{26}-3a^{23}-3a^{22}+4a^{20}-a^{19}-9a^{18}-3a^{17}+6a^{16}+9a^{15}-3a^{14}-4a^{13}+2a^{12}+7a^{11}+3a^{10}-a^{9}-3a^{8}-8a^{7}-a^{6}+4a^{5}+5a^{4}-13a^{3}-12a^{2}+5a+16$, $4a^{30}-7a^{29}+9a^{28}-6a^{27}+5a^{26}-2a^{25}-a^{24}-a^{22}+2a^{20}-4a^{16}+a^{15}-3a^{14}+4a^{13}-a^{12}+7a^{11}-6a^{10}+7a^{9}-9a^{8}+2a^{7}+2a^{6}-9a^{5}+18a^{4}-20a^{3}+18a^{2}-16a-2$, $6a^{30}-7a^{29}+a^{28}+2a^{27}-9a^{26}+10a^{25}+3a^{24}-6a^{23}+4a^{22}-7a^{21}-5a^{20}+15a^{19}-5a^{18}+5a^{16}-13a^{15}+3a^{14}+8a^{13}-8a^{12}+13a^{11}-a^{10}-17a^{9}+10a^{8}-a^{7}+a^{6}+17a^{5}-16a^{4}-9a^{3}+14a^{2}-11a+7$ (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: | \( 1630733111771600400 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{15}\cdot 1630733111771600400 \cdot 1}{2\cdot\sqrt{3514391585695303086629141906098469962323789296982843961159719}}\cr\approx \mathstrut & 0.816875441622716 \end{aligned}\] (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 $S_{31}$ 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 | $21{,}\,{\href{/padicField/2.7.0.1}{7} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ | R | $31$ | ${\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | $23{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | $30{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.11.0.1}{11} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}$ | $20{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.5.0.1}{5} }$ | $31$ | $30{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $26{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.13.0.1}{13} }{,}\,{\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $31$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.3.4.3 | $x^{3} + 6 x^{2} + 12$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
3.3.3.1 | $x^{3} + 6 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $[3/2]_{2}$ | |
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 | $[3/2, 3/2, 3/2, 3/2]_{2}^{4}$ | |
3.12.12.8 | $x^{12} + 42 x^{11} + 732 x^{10} + 6888 x^{9} + 37836 x^{8} + 101736 x^{7} + 148230 x^{6} + 129006 x^{5} + 72900 x^{4} + 28944 x^{3} + 8262 x^{2} + 1296 x + 81$ | $3$ | $4$ | $12$ | 12T119 | $[3/2, 3/2, 3/2]_{2}^{4}$ | |
\(571\) | $\Q_{571}$ | $x$ | $1$ | $1$ | $0$ | 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 $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
\(996\!\cdots\!887\) | $\Q_{99\!\cdots\!87}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{99\!\cdots\!87}$ | $x$ | $1$ | $1$ | $0$ | 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 $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |