Normalized defining polynomial
\( x^{29} - x - 2 \)
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: | \(689258003355570029305128342152413503002579071664128\) \(\medspace = 2^{28}\cdot 23\cdot 2505541501481\cdot 44556647066894154934970530051\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(56.63\) | 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: | \(2\), \(23\), \(2505541501481\), \(44556647066894154934970530051\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{25676\!\cdots\!27213}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$
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)
| |
Fundamental units: | $a^{15}-a-1$, $a^{4}-a^{2}+1$, $a^{8}-a^{6}+a^{4}-a^{2}+1$, $a^{27}+a^{25}+a^{23}+a^{21}+a^{19}+a^{17}+a^{15}+a^{13}+a^{11}+a^{9}+a^{7}+a^{5}+a^{4}+a^{3}+a^{2}+a+1$, $a^{22}+a^{15}+a^{8}+a+1$, $a^{20}-a^{10}+1$, $a^{28}-a^{27}+a^{26}-a^{24}+a^{23}-a^{22}+a^{20}-a^{19}+a^{18}-a^{16}+a^{15}-a^{14}+a^{13}-a^{11}+a^{10}-a^{9}+2a^{7}-2a^{6}+a^{5}-a^{3}+a^{2}-a-1$, $a^{20}+a^{19}+a^{18}+a^{17}+a^{16}+a^{15}+a^{14}+a^{13}+a^{12}+a^{11}-a^{9}-a^{8}-a^{7}-a^{6}-a^{5}-a^{4}-a^{3}-a^{2}-2a-1$, $a^{28}+a^{27}-a^{26}-a^{25}+a^{23}+a^{22}-a^{20}-a^{19}+a^{17}+a^{16}-2a^{14}-a^{13}+a^{11}+a^{10}-a^{9}-2a^{8}-a^{7}+a^{6}+2a^{5}+a^{4}-a^{3}-2a^{2}+1$, $3a^{27}-4a^{26}+3a^{25}-a^{24}-3a^{23}+4a^{22}-4a^{21}+2a^{20}+2a^{19}-3a^{18}+4a^{17}-2a^{16}-2a^{15}+4a^{14}-5a^{13}+3a^{12}-4a^{10}+5a^{9}-4a^{8}+a^{7}+3a^{6}-4a^{5}+5a^{4}-a^{3}-3a^{2}+5a-5$, $2a^{28}-3a^{27}+3a^{26}-a^{25}+4a^{24}-4a^{23}+2a^{22}-4a^{21}+3a^{20}-4a^{19}+4a^{18}-a^{17}+5a^{16}-4a^{15}+a^{14}-4a^{13}+2a^{12}-3a^{11}+2a^{10}+3a^{8}-2a^{7}-a^{6}+a^{4}-2a^{2}+a-3$, $5a^{28}-5a^{27}-6a^{26}-4a^{25}+7a^{24}+8a^{23}+4a^{22}-7a^{21}-8a^{20}-2a^{19}+9a^{18}+8a^{17}-11a^{15}-9a^{14}+2a^{13}+12a^{12}+10a^{11}-3a^{10}-12a^{9}-8a^{8}+8a^{7}+15a^{6}+9a^{5}-11a^{4}-16a^{3}-8a^{2}+13a+11$, $10a^{28}-4a^{27}-4a^{26}+8a^{25}-3a^{24}-8a^{23}+11a^{22}+3a^{21}-12a^{20}+4a^{19}+6a^{18}-9a^{17}+4a^{16}+8a^{15}-13a^{14}-2a^{13}+14a^{12}-7a^{11}-8a^{10}+12a^{9}-4a^{8}-10a^{7}+12a^{6}+a^{5}-15a^{4}+12a^{3}+7a^{2}-17a-7$, $6a^{28}+3a^{27}+4a^{26}-a^{25}-2a^{24}-7a^{23}-6a^{22}-7a^{21}-3a^{20}-2a^{19}+4a^{18}+6a^{17}+7a^{16}+5a^{15}+5a^{14}+a^{13}-2a^{12}-2a^{11}-2a^{10}-3a^{9}-3a^{8}-2a^{7}-6a^{6}-6a^{5}-5a^{4}-a^{3}+a^{2}+10a+7$ (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: | \( 261495167861816.16 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{14}\cdot 261495167861816.16 \cdot 1}{2\cdot\sqrt{689258003355570029305128342152413503002579071664128}}\cr\approx \mathstrut & 1.48864422233479 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 8841761993739701954543616000000 |
The 4565 conjugacy class representatives for $S_{29}$ are not computed |
Character table for $S_{29}$ 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 | R | $22{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | $15{,}\,{\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | $17{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | $29$ | $22{,}\,{\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | R | $29$ | $27{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $21{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | $27{,}\,{\href{/padicField/43.2.0.1}{2} }$ | $18{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $21{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.4.4.5 | $x^{4} + 2 x + 2$ | $4$ | $1$ | $4$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ | |
2.12.12.31 | $x^{12} + 2 x^{10} + 2 x^{9} + 6 x^{8} + 12 x^{7} + 32 x^{6} + 48 x^{5} + 76 x^{4} + 48 x^{3} + 40 x^{2} + 8 x + 8$ | $4$ | $3$ | $12$ | 12T205 | $[4/3, 4/3, 4/3, 4/3, 4/3, 4/3]_{3}^{6}$ | |
2.12.12.32 | $x^{12} - 4 x^{10} - 4 x^{9} + 26 x^{8} + 40 x^{7} - 4 x^{6} - 40 x^{5} + 28 x^{4} + 72 x^{3} + 24 x^{2} - 16 x + 8$ | $4$ | $3$ | $12$ | 12T129 | $[4/3, 4/3, 4/3, 4/3]_{3}^{6}$ | |
\(23\) | 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
23.5.0.1 | $x^{5} + 3 x + 18$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
23.20.0.1 | $x^{20} + x^{2} - 6 x + 20$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ | |
\(2505541501481\) | $\Q_{2505541501481}$ | $x$ | $1$ | $1$ | $0$ | 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 $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
\(445\!\cdots\!051\) | $\Q_{44\!\cdots\!51}$ | $x$ | $1$ | $1$ | $0$ | 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 $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ |