Normalized defining polynomial
\( x^{29} + 4x - 2 \)
Invariants
Degree: | $29$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 14]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(9553547283936279325927817224481489543638932340402393448448\) \(\medspace = 2^{28}\cdot 3\cdot 3783589506654278873\cdot 31\!\cdots\!07\) | 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: | \(99.84\) | 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: | $2^{28/29}3^{1/2}3783589506654278873^{1/2}3135447344393691704372001950407^{1/2}\approx 1.1649627540317457e+25$ | ||
Ramified primes: | \(2\), \(3\), \(3783589506654278873\), \(31354\!\cdots\!50407\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | $\Q(\sqrt{35589\!\cdots\!53933}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $14$ | 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: | $2a^{22}-4a^{15}+4a^{8}-1$, $a^{22}-a^{15}+2a-1$, $a^{28}-a^{27}+a^{25}-a^{24}+a^{22}-a^{21}+a^{19}-a^{18}+a^{16}-a^{15}+a^{13}-a^{12}+a^{10}-a^{9}+a^{7}-a^{6}+a^{4}-a^{3}+2a-1$, $4a^{28}-a^{27}+4a^{26}+2a^{25}-2a^{24}+3a^{23}-4a^{21}+a^{20}-a^{19}-7a^{18}+a^{17}-4a^{16}-11a^{15}+a^{14}-9a^{13}-11a^{12}-12a^{10}-12a^{9}-2a^{8}-12a^{7}-13a^{6}+2a^{5}-13a^{4}-13a^{3}+7a^{2}-16a+9$, $a^{28}+a^{27}+a^{26}-3a^{24}-a^{22}+3a^{21}+2a^{19}-5a^{18}+a^{17}-3a^{16}+6a^{15}-a^{14}+5a^{13}-7a^{12}+2a^{11}-5a^{10}+6a^{9}-a^{8}+6a^{7}-9a^{6}+2a^{5}-5a^{4}+4a^{3}+3a^{2}+2a-1$, $7a^{28}+a^{27}+7a^{25}+8a^{24}+a^{23}-3a^{22}+5a^{21}+8a^{20}+a^{19}-8a^{18}+8a^{16}+2a^{15}-11a^{14}-8a^{13}+5a^{12}+3a^{11}-13a^{10}-14a^{9}+a^{8}+5a^{7}-14a^{6}-23a^{5}-3a^{4}+8a^{3}-7a^{2}-30a+17$, $2a^{28}+2a^{27}+3a^{26}+a^{25}-2a^{24}-a^{23}+3a^{22}+3a^{21}-3a^{20}-7a^{19}-2a^{18}+6a^{17}+7a^{16}-6a^{14}-6a^{13}-2a^{12}+2a^{11}+5a^{10}+4a^{9}-a^{8}-6a^{7}-4a^{6}+2a^{5}+3a^{4}-2a^{3}-2a^{2}+4a+13$, $16a^{28}+22a^{27}+20a^{26}-11a^{25}-17a^{24}-5a^{23}+25a^{22}+14a^{21}-10a^{20}-31a^{19}+25a^{17}+27a^{16}-21a^{15}-32a^{14}-11a^{13}+41a^{12}+27a^{11}-15a^{10}-50a^{9}-9a^{8}+44a^{7}+39a^{6}-17a^{5}-64a^{4}-7a^{3}+42a^{2}+64a+25$, $9a^{28}+4a^{27}-2a^{26}-9a^{25}-8a^{24}-4a^{23}+5a^{22}+7a^{21}+11a^{20}+2a^{19}-3a^{18}-13a^{17}-7a^{16}-4a^{15}+9a^{14}+9a^{13}+14a^{12}-2a^{11}-8a^{10}-18a^{9}-8a^{8}-2a^{7}+14a^{6}+18a^{5}+17a^{4}-4a^{3}-17a^{2}-21a+19$, $27a^{28}-23a^{27}+17a^{26}-8a^{25}-3a^{24}+15a^{23}-27a^{22}+37a^{21}-43a^{20}+45a^{19}-40a^{18}+30a^{17}-15a^{16}-5a^{15}+26a^{14}-47a^{13}+66a^{12}-79a^{11}+85a^{10}-83a^{9}+69a^{8}-46a^{7}+13a^{6}+28a^{5}-70a^{4}+110a^{3}-141a^{2}+159a-53$, $10a^{28}+10a^{27}+7a^{26}+2a^{24}+a^{23}-5a^{22}+a^{21}+7a^{20}+a^{19}+7a^{18}+19a^{17}+7a^{16}-3a^{15}+11a^{14}+13a^{13}+3a^{12}+15a^{11}+24a^{10}+11a^{9}+13a^{8}+17a^{7}-3a^{6}+2a^{5}+29a^{4}+14a^{3}-3a^{2}+25a+61$, $43a^{28}+15a^{27}-12a^{26}+8a^{25}+19a^{24}-6a^{23}-16a^{22}+12a^{21}+14a^{20}-20a^{19}-6a^{18}+23a^{17}+4a^{16}-33a^{15}+6a^{14}+35a^{13}-12a^{12}-35a^{11}+17a^{10}+37a^{9}-32a^{8}-22a^{7}+35a^{6}+17a^{5}-51a^{4}+62a^{2}-20a+109$, $13a^{28}+17a^{27}+5a^{26}-10a^{25}-10a^{24}+7a^{23}+14a^{22}+4a^{21}-16a^{20}-13a^{19}+7a^{18}+21a^{17}+4a^{16}-17a^{15}-19a^{14}+9a^{13}+23a^{12}+7a^{11}-26a^{10}-21a^{9}+10a^{8}+34a^{7}+12a^{6}-27a^{5}-28a^{4}+14a^{3}+40a^{2}+11a+15$, $4a^{28}-4a^{27}+3a^{26}-4a^{25}+3a^{24}-3a^{23}+2a^{22}-a^{21}+a^{20}-a^{19}+a^{18}-a^{17}+2a^{16}-a^{15}+a^{14}-a^{13}+a^{12}-2a^{11}+a^{10}+2a^{9}-5a^{8}+7a^{7}-10a^{6}+12a^{5}-14a^{4}+19a^{3}-21a^{2}+25a-9$ (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: | \( 489687221548003600 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{14}\cdot 489687221548003600 \cdot 1}{2\cdot\sqrt{9553547283936279325927817224481489543638932340402393448448}}\cr\approx \mathstrut & 0.748780627007789 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 8841761993739701954543616000000 |
The 4565 conjugacy class representatives for $S_{29}$ |
Character table for $S_{29}$ |
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 | R | $15{,}\,{\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | $19{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | $22{,}\,{\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | $26{,}\,{\href{/padicField/19.3.0.1}{3} }$ | $18{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | $17{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }$ | $18{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | $23{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 $29$ | $29$ | $1$ | $28$ | |||
\(3\) | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.7.0.1 | $x^{7} + 2 x^{2} + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
3.8.0.1 | $x^{8} + 2 x^{5} + x^{4} + 2 x^{2} + 2 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
3.12.0.1 | $x^{12} + x^{6} + x^{5} + x^{4} + x^{2} + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
\(3783589506654278873\) | $\Q_{3783589506654278873}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{3783589506654278873}$ | $x$ | $1$ | $1$ | $0$ | 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 $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
\(313\!\cdots\!407\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $27$ | $1$ | $27$ | $0$ | $C_{27}$ | $[\ ]^{27}$ |