Normalized defining polynomial
\( x^{34} - 4x + 4 \)
Invariants
Degree: | $34$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 17]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-48276286010390714256067324090953417465726070579884714323083264\) \(\medspace = -\,2^{34}\cdot 41\cdot 131\cdot 149\cdot 2536801\cdot 13\!\cdots\!49\) | 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: | \(65.20\) | 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: | not computed | ||
Ramified primes: | \(2\), \(41\), \(131\), \(149\), \(2536801\), \(13841\!\cdots\!42649\) | 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: | $\Q(\sqrt{-28100\!\cdots\!11871}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2}a^{17}$, $\frac{1}{2}a^{18}$, $\frac{1}{2}a^{19}$, $\frac{1}{2}a^{20}$, $\frac{1}{2}a^{21}$, $\frac{1}{2}a^{22}$, $\frac{1}{2}a^{23}$, $\frac{1}{2}a^{24}$, $\frac{1}{2}a^{25}$, $\frac{1}{2}a^{26}$, $\frac{1}{2}a^{27}$, $\frac{1}{2}a^{28}$, $\frac{1}{2}a^{29}$, $\frac{1}{2}a^{30}$, $\frac{1}{2}a^{31}$, $\frac{1}{2}a^{32}$, $\frac{1}{2}a^{33}$
Monogenic: | Not computed | |
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: | $\frac{1}{2}a^{17}$, $\frac{1}{2}a^{32}+\frac{1}{2}a^{30}-\frac{1}{2}a^{22}+\frac{1}{2}a^{21}-\frac{1}{2}a^{20}+\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-a^{11}+a^{10}-a^{9}+a^{8}-a^{7}-1$, $\frac{1}{2}a^{27}+\frac{1}{2}a^{26}-\frac{1}{2}a^{24}-\frac{1}{2}a^{23}-\frac{1}{2}a^{22}+\frac{1}{2}a^{20}+\frac{1}{2}a^{19}-\frac{1}{2}a^{17}-a^{11}-a^{10}+a^{8}+a^{7}-a^{4}-a^{3}+a$, $\frac{1}{2}a^{28}+\frac{1}{2}a^{27}+\frac{1}{2}a^{24}+\frac{1}{2}a^{23}-\frac{1}{2}a^{21}+\frac{1}{2}a^{19}-\frac{1}{2}a^{17}$, $a^{33}-\frac{1}{2}a^{30}-\frac{1}{2}a^{28}-\frac{1}{2}a^{27}-a^{26}-\frac{1}{2}a^{25}+\frac{1}{2}a^{23}+\frac{1}{2}a^{22}+\frac{1}{2}a^{20}+\frac{1}{2}a^{19}+\frac{3}{2}a^{18}-2a^{15}-a^{13}+a^{12}-2a^{11}-2a^{9}+2a^{8}+3a^{6}-a^{5}+2a^{4}-a^{3}+3a^{2}-a-3$, $\frac{1}{2}a^{33}+\frac{1}{2}a^{32}-\frac{1}{2}a^{28}-\frac{1}{2}a^{26}-\frac{1}{2}a^{25}-\frac{1}{2}a^{24}-\frac{3}{2}a^{23}-\frac{1}{2}a^{22}-a^{21}-\frac{1}{2}a^{20}-\frac{1}{2}a^{18}+\frac{1}{2}a^{17}+a^{12}+a^{11}+a^{10}+a^{9}+a^{7}+a^{4}+a^{2}-4$, $\frac{9}{2}a^{33}+\frac{9}{2}a^{32}+4a^{31}+\frac{9}{2}a^{30}+\frac{9}{2}a^{29}+3a^{28}+3a^{27}+\frac{7}{2}a^{26}+2a^{25}+3a^{24}+\frac{7}{2}a^{23}+3a^{22}+\frac{7}{2}a^{21}+\frac{7}{2}a^{20}+3a^{19}+2a^{18}+2a^{17}+a^{16}+a^{15}+a^{14}+a^{13}+3a^{12}+a^{11}+2a^{10}+3a^{9}+a^{6}-a^{5}-3a^{4}+3a^{3}-2a-13$, $a^{30}+a^{29}+\frac{1}{2}a^{28}+a^{27}+a^{26}+\frac{3}{2}a^{25}+\frac{1}{2}a^{23}+\frac{1}{2}a^{22}-\frac{1}{2}a^{20}-\frac{1}{2}a^{19}-\frac{1}{2}a^{17}-a^{16}-a^{15}-a^{13}-2a^{11}-a^{6}+2a^{5}+2a$, $2a^{33}+\frac{3}{2}a^{32}+a^{31}+2a^{29}+\frac{3}{2}a^{28}-\frac{1}{2}a^{27}+a^{26}+\frac{3}{2}a^{25}+a^{24}-a^{23}+\frac{3}{2}a^{22}+\frac{3}{2}a^{21}-\frac{1}{2}a^{20}-\frac{1}{2}a^{19}+\frac{3}{2}a^{18}+\frac{1}{2}a^{17}-a^{16}+2a^{14}-a^{13}-a^{12}+2a^{11}-a^{9}-2a^{8}+5a^{7}-3a^{6}-2a^{5}+a^{4}+3a^{3}-2a^{2}-6a$, $\frac{3}{2}a^{32}-\frac{1}{2}a^{31}+a^{30}-\frac{1}{2}a^{28}+\frac{1}{2}a^{27}-2a^{26}+\frac{1}{2}a^{25}-\frac{5}{2}a^{24}-\frac{1}{2}a^{23}-a^{22}-2a^{21}+\frac{1}{2}a^{20}-3a^{19}+\frac{3}{2}a^{18}-\frac{3}{2}a^{17}+a^{16}+a^{15}-a^{14}+3a^{13}-2a^{12}+4a^{11}-a^{10}+2a^{9}+2a^{8}-2a^{7}+4a^{6}-5a^{5}+5a^{4}-2a^{3}+a^{2}+2a-6$, $3a^{33}-a^{32}-2a^{31}+\frac{7}{2}a^{30}+\frac{5}{2}a^{29}-a^{28}-\frac{1}{2}a^{27}+\frac{3}{2}a^{26}+\frac{7}{2}a^{25}-\frac{1}{2}a^{24}-\frac{5}{2}a^{23}+a^{22}+3a^{21}+\frac{1}{2}a^{20}-\frac{13}{2}a^{19}-\frac{1}{2}a^{18}+\frac{13}{2}a^{17}-4a^{16}-7a^{15}+a^{14}+5a^{13}-2a^{12}-8a^{11}+3a^{10}+5a^{9}-a^{8}-3a^{7}-4a^{6}+10a^{5}+5a^{4}-12a^{3}-a^{2}+12a-6$, $\frac{1}{2}a^{31}+a^{30}+\frac{1}{2}a^{29}-\frac{1}{2}a^{28}-a^{27}-\frac{3}{2}a^{26}-\frac{3}{2}a^{25}+\frac{1}{2}a^{23}-\frac{1}{2}a^{22}-\frac{1}{2}a^{21}-\frac{1}{2}a^{20}-\frac{1}{2}a^{19}+a^{18}+2a^{17}+a^{16}-a^{14}-2a^{13}-a^{12}+a^{11}-a^{9}-a^{8}-2a^{7}-a^{6}+2a^{5}+3a^{4}+2a^{3}-a-1$, $\frac{5}{2}a^{32}-\frac{1}{2}a^{31}+a^{30}-\frac{3}{2}a^{29}+2a^{28}+\frac{3}{2}a^{26}-2a^{25}-a^{23}+\frac{3}{2}a^{22}-\frac{3}{2}a^{19}-\frac{3}{2}a^{18}+a^{15}-3a^{14}-2a^{12}+4a^{11}-3a^{10}+a^{9}-7a^{8}+4a^{7}-4a^{6}+7a^{5}-7a^{4}+5a^{3}-10a^{2}+8a-7$, $\frac{9}{2}a^{33}+\frac{9}{2}a^{32}+4a^{31}+4a^{30}+\frac{7}{2}a^{29}+\frac{7}{2}a^{28}+4a^{27}+\frac{7}{2}a^{26}+3a^{25}+\frac{5}{2}a^{24}+\frac{5}{2}a^{23}+3a^{22}+3a^{21}+2a^{20}+2a^{19}+a^{18}+2a^{17}+2a^{16}+2a^{15}+a^{13}+a^{12}+a^{11}+a^{10}+a^{9}-2a^{8}+a^{7}+a^{6}-2a^{2}-17$, $\frac{5}{2}a^{33}+3a^{32}+5a^{31}+\frac{9}{2}a^{30}+2a^{29}+2a^{28}+4a^{27}+\frac{7}{2}a^{26}+\frac{1}{2}a^{25}+\frac{1}{2}a^{24}+\frac{7}{2}a^{23}+\frac{7}{2}a^{22}-\frac{1}{2}a^{21}-a^{20}+3a^{19}+4a^{18}-a^{16}+3a^{15}+4a^{14}-a^{13}-3a^{12}+2a^{11}+4a^{10}-2a^{9}-5a^{8}+a^{7}+4a^{6}-2a^{5}-5a^{4}+2a^{3}+6a^{2}-2a-17$, $\frac{1}{2}a^{33}-\frac{1}{2}a^{32}-a^{31}+a^{30}-a^{29}-\frac{1}{2}a^{28}-\frac{1}{2}a^{27}-\frac{3}{2}a^{26}+\frac{1}{2}a^{25}+a^{23}-\frac{1}{2}a^{21}+a^{20}-2a^{19}+a^{18}-\frac{1}{2}a^{17}-3a^{16}+3a^{15}-a^{14}+a^{13}+3a^{12}-3a^{11}+a^{10}-a^{9}+a^{7}-2a^{6}+2a^{5}-3a^{4}+3a^{3}+3a^{2}-5a+4$ (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: | \( 380534576916766400 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{17}\cdot 380534576916766400 \cdot 1}{2\cdot\sqrt{48276286010390714256067324090953417465726070579884714323083264}}\cr\approx \mathstrut & 1.01520784544478 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 295232799039604140847618609643520000000 |
The 12310 conjugacy class representatives for $S_{34}$ are not computed |
Character table for $S_{34}$ |
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 | $28{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $31{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | $24{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $32{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.8.0.1}{8} }^{2}{,}\,{\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.10.0.1}{10} }^{2}{,}\,{\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $23{,}\,{\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | R | $31{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | $25{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | $28{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 $34$ | $34$ | $1$ | $34$ | |||
\(41\) | $\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
41.2.1.2 | $x^{2} + 123$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
41.3.0.1 | $x^{3} + x + 35$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
41.6.0.1 | $x^{6} + 4 x^{4} + 33 x^{3} + 39 x^{2} + 6 x + 6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
41.20.0.1 | $x^{20} - x + 6$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ | |
\(131\) | 131.2.1.2 | $x^{2} + 131$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
131.4.0.1 | $x^{4} + 9 x^{2} + 109 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
131.4.0.1 | $x^{4} + 9 x^{2} + 109 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
131.6.0.1 | $x^{6} + 2 x^{4} + 66 x^{3} + 4 x^{2} + 22 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
131.7.0.1 | $x^{7} + 10 x + 129$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
131.11.0.1 | $x^{11} + 6 x + 129$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | |
\(149\) | $\Q_{149}$ | $x + 147$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
149.2.0.1 | $x^{2} + 145 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
149.2.1.1 | $x^{2} + 149$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
149.3.0.1 | $x^{3} + 3 x + 147$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
149.6.0.1 | $x^{6} + x^{4} + 105 x^{3} + 33 x^{2} + 55 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
149.7.0.1 | $x^{7} + 19 x + 147$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
149.13.0.1 | $x^{13} + 4 x + 147$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | |
\(2536801\) | $\Q_{2536801}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2536801}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | ||
\(138\!\cdots\!649\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $32$ | $1$ | $32$ | $0$ | 32T33 | $[\ ]^{32}$ |