Normalized defining polynomial
\( x^{31} - 4x - 2 \)
Invariants
Degree: | $31$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[3, 14]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(949505236871193060019419348344715648164966369654411826462457856\) \(\medspace = 2^{30}\cdot 2383\cdot 37\!\cdots\!43\) | 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: | \(107.53\) | 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: | $2^{30/31}2383^{1/2}371085044823230927769982392525957604255428134265743^{1/2}\approx 1.839154007745496e+27$ | ||
Ramified primes: | \(2\), \(2383\), \(37108\!\cdots\!65743\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{88429\!\cdots\!65569}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $16$ | 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: | $a+1$, $2a^{16}-4a-1$, $a^{15}-a^{14}+a^{13}-a^{12}+a^{11}-a^{10}+a^{9}-a^{8}+a^{7}-a^{6}+a^{5}-a^{4}+a^{3}-a^{2}+a+1$, $3a^{30}-a^{29}-2a^{28}+2a^{27}-2a^{25}+2a^{24}-a^{21}+a^{20}+a^{19}-3a^{18}+4a^{16}-5a^{15}+a^{14}+5a^{13}-4a^{12}-a^{11}+4a^{10}-2a^{9}-2a^{8}+2a^{6}-a^{5}-3a^{4}+6a^{3}+a^{2}-8a-5$, $8a^{30}-6a^{29}+3a^{28}+2a^{27}-2a^{25}+3a^{24}+2a^{23}-4a^{22}-4a^{21}+3a^{20}+2a^{19}-4a^{18}-2a^{17}+4a^{16}-5a^{14}+2a^{13}+8a^{12}-6a^{10}+3a^{9}+7a^{8}-3a^{7}-6a^{6}+4a^{5}+a^{4}-13a^{3}-8a^{2}+10a-25$, $9a^{30}+9a^{29}+10a^{28}+10a^{27}+9a^{26}+9a^{25}+8a^{24}+5a^{23}+2a^{22}-3a^{20}-8a^{19}-11a^{18}-14a^{17}-19a^{16}-21a^{15}-22a^{14}-25a^{13}-27a^{12}-25a^{11}-21a^{10}-20a^{9}-15a^{8}-8a^{7}-3a^{6}+6a^{5}+16a^{4}+24a^{3}+31a^{2}+41a+15$, $a^{30}+a^{29}-a^{28}-3a^{27}+8a^{26}-6a^{25}-3a^{24}+11a^{23}-10a^{22}+2a^{21}+6a^{20}-8a^{19}+6a^{18}-a^{17}-3a^{16}+2a^{15}+2a^{14}-2a^{13}-7a^{12}+14a^{11}-8a^{10}-10a^{9}+21a^{8}-15a^{7}-2a^{6}+13a^{5}-14a^{4}+7a^{3}+a^{2}-7a-1$, $30a^{30}-15a^{29}+11a^{28}+2a^{27}-a^{26}-3a^{25}+2a^{24}-7a^{21}-4a^{20}+7a^{19}-3a^{17}+3a^{16}+5a^{15}+11a^{14}+3a^{13}-6a^{12}+9a^{11}+8a^{10}-6a^{9}-7a^{8}-8a^{7}+3a^{6}+4a^{5}-16a^{4}-4a^{3}+15a^{2}+11a-113$, $4a^{30}+5a^{29}-2a^{28}+4a^{27}+4a^{26}-3a^{25}+a^{24}+3a^{23}-7a^{22}+a^{20}-7a^{19}+4a^{17}-7a^{16}+a^{15}+5a^{14}-9a^{13}-a^{12}+3a^{11}-11a^{10}-5a^{9}+3a^{8}-14a^{7}-3a^{6}+6a^{5}-8a^{4}+5a^{3}+19a^{2}+a-3$, $3a^{30}-3a^{29}+14a^{28}-23a^{27}+26a^{26}-29a^{25}+19a^{24}-5a^{23}+10a^{21}-26a^{20}+29a^{19}-32a^{18}+31a^{17}-11a^{16}+2a^{15}+6a^{14}-29a^{13}+40a^{12}-37a^{11}+38a^{10}-22a^{9}-2a^{7}-21a^{6}+51a^{5}-50a^{4}+50a^{3}-37a^{2}+a-5$, $13a^{30}+a^{29}-16a^{28}+18a^{27}-12a^{26}-9a^{25}+18a^{24}-22a^{23}+5a^{22}+14a^{21}-23a^{20}+21a^{19}+3a^{18}-17a^{17}+31a^{16}-14a^{15}-7a^{14}+27a^{13}-33a^{12}+8a^{11}+14a^{10}-42a^{9}+27a^{8}-3a^{7}-31a^{6}+48a^{5}-21a^{4}-7a^{3}+53a^{2}-41a-31$, $8a^{30}-4a^{29}-9a^{28}+9a^{27}-3a^{26}-3a^{25}+14a^{24}-6a^{23}-12a^{22}+10a^{21}-3a^{20}-4a^{19}+18a^{18}-6a^{17}-16a^{16}+12a^{15}-2a^{14}-6a^{13}+23a^{12}-5a^{11}-23a^{10}+13a^{9}-9a^{7}+29a^{6}-3a^{5}-32a^{4}+15a^{3}+2a^{2}-13a+7$, $9a^{30}-18a^{29}+21a^{28}-8a^{27}+5a^{26}-7a^{25}-10a^{24}+21a^{23}-13a^{22}+17a^{21}-26a^{20}+9a^{19}+4a^{18}+2a^{17}+13a^{16}-32a^{15}+19a^{14}-13a^{13}+28a^{12}-8a^{11}-21a^{10}+9a^{9}-13a^{8}+45a^{7}-32a^{6}+3a^{5}-20a^{4}+11a^{3}+37a^{2}-30a-21$, $3a^{30}-7a^{29}+6a^{28}-3a^{27}+a^{26}+a^{25}-4a^{24}+8a^{22}-9a^{21}+5a^{20}-2a^{19}-a^{18}+4a^{17}-3a^{16}-5a^{15}+14a^{14}-9a^{13}+a^{12}-a^{11}-3a^{10}+7a^{9}+a^{8}-16a^{7}+16a^{6}-3a^{5}-10a^{2}+6a+5$, $14a^{30}+11a^{29}+6a^{28}+3a^{26}-10a^{25}-18a^{24}-18a^{23}+7a^{22}+19a^{21}+16a^{20}-2a^{19}+4a^{18}+4a^{17}-6a^{16}-25a^{15}-7a^{14}+11a^{13}+13a^{12}-14a^{11}-12a^{10}+8a^{9}+31a^{8}+11a^{7}-2a^{6}-2a^{5}+6a^{4}-36a^{3}-62a^{2}-32a-5$, $7a^{30}-2a^{29}+2a^{28}+a^{27}-2a^{26}-3a^{25}-2a^{24}+a^{23}+7a^{22}+3a^{21}-2a^{20}-6a^{19}-3a^{18}-a^{17}+3a^{16}+4a^{15}+a^{14}+4a^{13}+a^{12}-2a^{11}-11a^{10}-8a^{9}+5a^{8}+11a^{7}+12a^{6}-9a^{4}-13a^{3}+5a-27$ (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: | \( 36948458623552070000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{14}\cdot 36948458623552070000 \cdot 1}{2\cdot\sqrt{949505236871193060019419348344715648164966369654411826462457856}}\cr\approx \mathstrut & 0.716845877209198 \end{aligned}\] (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 | $29{,}\,{\href{/padicField/3.2.0.1}{2} }$ | $24{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $25{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | $17{,}\,{\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | $22{,}\,{\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | $30{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.5.0.1}{5} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.5.0.1}{5} }$ | $28{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | $26{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 $31$ | $31$ | $1$ | $30$ | |||
\(2383\) | $\Q_{2383}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2383}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{2383}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
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 $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
\(371\!\cdots\!743\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |