Normalized defining polynomial
\( x^{31} - 4x - 1 \)
Invariants
Degree: | $31$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[3, 14]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(949505255199079208246627035002460041389356970940685243955265569\) \(\medspace = 24365060088671\cdot 38\!\cdots\!39\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
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))
| |
Galois root discriminant: | $24365060088671^{1/2}38969953357125920236474466375786993070484887309439^{1/2}\approx 3.0814043149172737e+31$ | ||
Ramified primes: | \(24365060088671\), \(38969\!\cdots\!09439\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{94950\!\cdots\!65569}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $16$ | 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: | $a^{30}-4$, $a^{16}+2a$, $a^{29}+a^{28}+a^{27}+a^{26}+a^{25}+a^{24}+a^{23}+a^{22}+a^{21}+a^{20}+a^{19}+a^{18}+a^{17}+a^{16}+a^{15}+a^{14}+a^{13}+a^{12}+a^{11}+a^{10}+a^{9}+a^{8}+a^{7}+a^{6}+a^{5}+a^{4}+a^{3}+a^{2}+a+2$, $6a^{30}+a^{29}+2a^{28}+2a^{27}-3a^{26}+4a^{25}-7a^{24}-7a^{22}+a^{21}+5a^{20}+2a^{19}+8a^{18}-5a^{17}+6a^{16}-5a^{15}-a^{14}-7a^{13}-10a^{12}+5a^{11}-a^{10}+17a^{9}-2a^{8}+7a^{7}-a^{6}-3a^{5}+a^{4}-22a^{3}-13a-5$, $2a^{30}-10a^{29}+5a^{28}+9a^{27}-11a^{26}-a^{25}+13a^{24}-8a^{23}-8a^{22}+12a^{21}-14a^{19}+5a^{18}+11a^{17}-13a^{16}-6a^{15}+20a^{14}-6a^{13}-17a^{12}+22a^{11}+6a^{10}-26a^{9}+12a^{8}+17a^{7}-24a^{6}-7a^{5}+24a^{4}-10a^{3}-25a^{2}+23a+8$, $14a^{30}-18a^{28}-37a^{27}-49a^{26}-50a^{25}-42a^{24}-29a^{23}-13a^{22}+12a^{21}+40a^{20}+65a^{19}+76a^{18}+70a^{17}+55a^{16}+33a^{15}+4a^{14}-37a^{13}-78a^{12}-107a^{11}-112a^{10}-95a^{9}-69a^{8}-31a^{7}+20a^{6}+84a^{5}+139a^{4}+167a^{3}+159a^{2}+125a+25$, $2a^{30}+2a^{29}-a^{28}+4a^{27}+4a^{24}+a^{23}-4a^{22}+8a^{21}-6a^{20}+4a^{19}-2a^{18}-2a^{17}+a^{16}-2a^{15}-8a^{14}+5a^{13}-6a^{12}-7a^{11}+5a^{10}-11a^{9}+4a^{8}-6a^{7}-3a^{6}+3a^{5}+3a^{4}-11a^{3}+18a^{2}-6a-4$, $a^{30}-3a^{29}+2a^{28}-3a^{27}+a^{25}-a^{24}-2a^{23}+7a^{22}-8a^{21}+8a^{20}-2a^{18}+7a^{17}-2a^{16}+5a^{15}-a^{14}+5a^{13}-a^{12}+a^{11}+4a^{10}-2a^{9}-6a^{8}+13a^{7}-19a^{6}+10a^{5}-6a^{4}-10a^{3}+6a^{2}-12a-3$, $12a^{30}-a^{29}+8a^{28}-8a^{27}+2a^{26}-15a^{25}-4a^{24}-15a^{23}-5a^{22}-8a^{21}+a^{20}+6a^{19}+6a^{18}+19a^{17}+9a^{16}+23a^{15}+a^{14}+17a^{13}-15a^{12}+a^{11}-28a^{10}-11a^{9}-32a^{8}-12a^{7}-16a^{6}-2a^{5}+12a^{4}+17a^{3}+39a^{2}+21a+1$, $4a^{30}-5a^{29}+4a^{28}-3a^{27}+3a^{26}-a^{25}+a^{24}+a^{23}-3a^{22}+4a^{21}-7a^{20}+8a^{19}-9a^{18}+12a^{17}-10a^{16}+15a^{15}-13a^{14}+14a^{13}-14a^{12}+9a^{11}-12a^{10}+7a^{9}-7a^{8}+5a^{7}+3a^{6}+9a^{4}-5a^{3}+9a^{2}-13a-5$, $20a^{30}+2a^{29}-28a^{28}+37a^{27}-12a^{26}-30a^{25}+50a^{24}-32a^{23}-19a^{22}+58a^{21}-51a^{20}+3a^{19}+54a^{18}-64a^{17}+22a^{16}+36a^{15}-63a^{14}+33a^{13}+14a^{12}-42a^{11}+35a^{10}-5a^{9}-11a^{8}+16a^{7}-13a^{6}+15a^{5}-26a^{4}+8a^{3}+31a^{2}-72a-19$, $45a^{30}+118a^{29}-77a^{28}-116a^{27}+106a^{26}+106a^{25}-135a^{24}-85a^{23}+172a^{22}+65a^{21}-195a^{20}-27a^{19}+223a^{18}-15a^{17}-244a^{16}+62a^{15}+247a^{14}-126a^{13}-255a^{12}+182a^{11}+238a^{10}-249a^{9}-207a^{8}+321a^{7}+172a^{6}-380a^{5}-105a^{4}+441a^{3}+24a^{2}-494a-117$, $2a^{30}-a^{29}-9a^{26}-5a^{25}+6a^{24}-3a^{23}-6a^{22}-a^{21}-9a^{20}+2a^{19}+11a^{18}-7a^{17}-7a^{16}+5a^{15}+4a^{14}+9a^{13}+10a^{12}-9a^{11}-2a^{10}+24a^{9}+9a^{8}-2a^{7}+6a^{6}-8a^{5}+10a^{4}+25a^{3}-14a^{2}-19a-4$, $7a^{30}+44a^{29}+30a^{28}-2a^{27}-55a^{26}-31a^{25}-a^{24}+65a^{23}+36a^{22}-73a^{20}-48a^{19}+4a^{18}+76a^{17}+65a^{16}-12a^{15}-77a^{14}-85a^{13}+22a^{12}+82a^{11}+108a^{10}-27a^{9}-92a^{8}-128a^{7}+24a^{6}+110a^{5}+140a^{4}-16a^{3}-142a^{2}-151a-28$, $86a^{30}-6a^{29}-89a^{28}+63a^{27}+54a^{26}-102a^{25}+6a^{24}+108a^{23}-81a^{22}-70a^{21}+129a^{20}-a^{19}-140a^{18}+90a^{17}+94a^{16}-153a^{15}-2a^{14}+178a^{13}-112a^{12}-127a^{11}+198a^{10}+18a^{9}-229a^{8}+120a^{7}+157a^{6}-233a^{5}-22a^{4}+278a^{3}-155a^{2}-210a-40$, $59a^{30}-138a^{29}-253a^{28}-25a^{27}+281a^{26}+279a^{25}-20a^{24}-232a^{23}-120a^{22}+21a^{21}-124a^{20}-153a^{19}+279a^{18}+543a^{17}+43a^{16}-533a^{15}-393a^{14}+73a^{13}+167a^{12}+39a^{11}+155a^{10}+416a^{9}+233a^{8}-543a^{7}-933a^{6}-142a^{5}+822a^{4}+571a^{3}-135a^{2}+2a+14$ (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: | \( 38197936820101820000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{14}\cdot 38197936820101820000 \cdot 1}{2\cdot\sqrt{949505255199079208246627035002460041389356970940685243955265569}}\cr\approx \mathstrut & 0.741087295199414 \end{aligned}\] (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 | ${\href{/padicField/2.5.0.1}{5} }^{6}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | $29{,}\,{\href{/padicField/3.2.0.1}{2} }$ | $31$ | $17{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $25{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $30{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $26{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.6.0.1}{6} }$ | $30{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.5.0.1}{5} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $29{,}\,{\href{/padicField/37.2.0.1}{2} }$ | $25{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.5.0.1}{5} }$ | $19{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(24365060088671\) | $\Q_{24365060088671}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
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 $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
\(389\!\cdots\!439\) | $\Q_{38\!\cdots\!39}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{38\!\cdots\!39}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
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 $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |