Normalized defining polynomial
\( x^{31} + 3x - 1 \)
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)
| |
Discriminant: | \(-127173474825665679717014022838958610643029059314756044734431\) \(\medspace = -\,43\cdot 39047\cdot 455317\cdot 317197259\cdot 45044106017368643\cdot 11642845821394330173359\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(80.65\) | 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: | $43^{1/2}39047^{1/2}455317^{1/2}317197259^{1/2}45044106017368643^{1/2}11642845821394330173359^{1/2}\approx 3.566139016158311e+29$ | ||
Ramified primes: | \(43\), \(39047\), \(455317\), \(317197259\), \(45044106017368643\), \(11642845821394330173359\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-12717\!\cdots\!34431}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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)
| |
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: | $a$, $a^{29}-a^{27}-a^{22}+a^{20}-a^{19}-a^{18}+a^{15}-2a^{13}+a^{11}+a^{9}-a^{8}-a^{7}+2a^{6}-a^{4}+a^{3}+a+1$, $4a^{29}-5a^{28}+3a^{27}+3a^{26}-7a^{25}+4a^{24}+a^{23}-7a^{22}+6a^{21}-2a^{20}-6a^{19}+9a^{18}-4a^{17}-4a^{16}+10a^{15}-7a^{14}-a^{13}+9a^{12}-12a^{11}+2a^{10}+8a^{9}-14a^{8}+7a^{7}+6a^{6}-13a^{5}+14a^{4}+a^{3}-15a^{2}+17a-5$, $a^{30}+a^{29}-a^{26}+a^{25}-a^{22}+a^{20}-a^{19}+a^{18}-2a^{17}+2a^{16}-a^{13}+2a^{10}-2a^{9}+a^{8}-2a^{7}+4a^{6}-3a^{5}+2a^{4}-3a^{3}+2a^{2}+a+1$, $a^{30}-2a^{29}-3a^{28}-2a^{27}-a^{26}+a^{24}+3a^{23}+3a^{22}-a^{21}-3a^{20}-2a^{19}-2a^{18}-2a^{17}+a^{16}+4a^{15}+5a^{14}+a^{13}-3a^{12}-2a^{11}-3a^{10}-4a^{9}+6a^{7}+6a^{6}+3a^{5}-2a^{4}-2a^{3}-3a^{2}-6a+1$, $2a^{30}+a^{29}-5a^{28}-3a^{27}-3a^{26}+5a^{25}+4a^{24}+5a^{23}-4a^{22}-4a^{21}-7a^{20}+3a^{19}+3a^{18}+8a^{17}-a^{16}-3a^{15}-9a^{14}-2a^{13}+2a^{12}+10a^{11}+5a^{10}-9a^{8}-9a^{7}-a^{6}+6a^{5}+11a^{4}+2a^{3}-3a^{2}-12a+3$, $2a^{30}-2a^{29}+2a^{28}+2a^{27}+2a^{26}-3a^{25}+4a^{24}-a^{23}-a^{22}-4a^{21}+4a^{20}-5a^{19}-a^{18}-2a^{17}+4a^{16}-7a^{15}+2a^{14}+a^{13}+2a^{12}-5a^{11}+7a^{10}+3a^{9}-a^{7}+7a^{6}-a^{5}-5a^{4}+4a^{3}+a^{2}-5a$, $3a^{30}-a^{29}+2a^{27}-a^{26}+a^{25}+a^{24}-3a^{23}+2a^{22}-2a^{21}-a^{20}+4a^{19}-5a^{18}+6a^{17}-4a^{16}+2a^{15}-2a^{12}+4a^{11}-9a^{10}+8a^{9}-9a^{8}+6a^{7}-2a^{6}+2a^{5}-2a^{4}+5a^{3}-10a^{2}+11a-2$, $a^{30}+2a^{29}-a^{28}+a^{26}-a^{25}+2a^{24}+a^{23}-a^{22}-a^{21}-a^{20}+5a^{19}+2a^{18}-6a^{17}-a^{16}+3a^{15}+5a^{14}-2a^{13}-8a^{12}+4a^{11}+5a^{10}-2a^{9}-3a^{8}-3a^{7}+6a^{6}-7a^{4}+3a^{3}+a^{2}+1$, $2a^{30}+5a^{29}-a^{28}-8a^{27}+3a^{26}+6a^{25}-2a^{24}-8a^{23}+3a^{22}+8a^{21}-4a^{20}-7a^{19}+a^{18}+11a^{17}-5a^{16}-9a^{15}+3a^{14}+12a^{13}-5a^{12}-12a^{11}+7a^{10}+11a^{9}-3a^{8}-17a^{7}+10a^{6}+13a^{5}-7a^{4}-17a^{3}+11a^{2}+17a-7$, $2a^{30}+a^{29}-a^{27}-a^{26}-a^{25}+2a^{24}+2a^{23}-a^{21}-2a^{20}-a^{19}+2a^{17}+2a^{16}-a^{15}-2a^{14}-2a^{13}-a^{12}+a^{11}+3a^{10}+3a^{9}-2a^{8}-2a^{7}-a^{6}+3a^{4}+3a^{3}+2a^{2}-5a+2$, $28a^{30}+14a^{29}-16a^{28}-a^{27}+30a^{26}+11a^{25}-27a^{24}-10a^{23}+26a^{22}+a^{21}-42a^{20}-16a^{19}+27a^{18}-2a^{17}-45a^{16}-10a^{15}+40a^{14}+9a^{13}-38a^{12}+3a^{11}+60a^{10}+19a^{9}-41a^{8}+5a^{7}+66a^{6}+8a^{5}-64a^{4}-7a^{3}+58a^{2}-13a-2$, $4a^{30}+4a^{29}-8a^{28}+4a^{27}-16a^{26}+20a^{25}-10a^{24}-a^{23}+17a^{22}-14a^{21}+11a^{20}-21a^{19}+16a^{18}-7a^{17}-15a^{16}+33a^{15}-20a^{14}+17a^{13}-16a^{12}+8a^{11}+3a^{10}-38a^{9}+46a^{8}-25a^{7}+13a^{6}+2a^{5}-3a^{4}+24a^{3}-63a^{2}+52a-12$, $2a^{30}-a^{29}-3a^{28}+2a^{26}-a^{25}-3a^{24}+a^{23}+2a^{22}-2a^{21}-a^{20}+3a^{19}+a^{18}-a^{17}-2a^{15}-a^{14}+3a^{13}-a^{12}-4a^{11}+5a^{10}+4a^{9}-7a^{8}-a^{7}+11a^{6}-9a^{4}+2a^{3}+7a^{2}-5a$, $a^{30}-9a^{29}+8a^{28}+7a^{26}-6a^{25}-7a^{24}+2a^{23}+a^{22}+12a^{21}-8a^{20}+2a^{19}-14a^{18}+10a^{17}+3a^{16}+4a^{15}-3a^{14}-11a^{13}+7a^{12}-5a^{11}+20a^{10}-18a^{9}+5a^{8}-13a^{7}+14a^{6}+4a^{5}-2a^{4}-2a^{3}-23a^{2}+26a-7$ (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: | \( 253630612977039870 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{15}\cdot 253630612977039870 \cdot 1}{2\cdot\sqrt{127173474825665679717014022838958610643029059314756044734431}}\cr\approx \mathstrut & 0.667884597016347 \end{aligned}\] (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 | $21{,}\,{\href{/padicField/2.7.0.1}{7} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ | $30{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.9.0.1}{9} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $26{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | $16{,}\,15$ | $15{,}\,{\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | $26{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $15^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.5.0.1}{5} }$ | $22{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | R | $24{,}\,{\href{/padicField/47.7.0.1}{7} }$ | $31$ | $17{,}\,{\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(43\) | 43.2.1.1 | $x^{2} + 86$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
43.4.0.1 | $x^{4} + 5 x^{2} + 42 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
43.5.0.1 | $x^{5} + 8 x + 40$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
43.20.0.1 | $x^{20} - x + 18$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ | |
\(39047\) | 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 $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | ||
\(455317\) | $\Q_{455317}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
\(317197259\) | $\Q_{317197259}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $24$ | $1$ | $24$ | $0$ | $C_{24}$ | $[\ ]^{24}$ | ||
\(45044106017368643\) | $\Q_{45044106017368643}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{45044106017368643}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{45044106017368643}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{45044106017368643}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
\(116\!\cdots\!359\) | $\Q_{11\!\cdots\!59}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{11\!\cdots\!59}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{11\!\cdots\!59}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ |