Normalized defining polynomial
\( x^{35} + 2x - 1 \)
Invariants
Degree: | $35$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 17]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-403955037978023056104591172895949491439236366378374168898704123\) \(\medspace = -\,256048319\cdot 15\!\cdots\!17\) | 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: | \(61.48\) | 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: | $256048319^{1/2}1577651591526453474215509975271306082814925125043973317^{1/2}\approx 2.009863273902041e+31$ | ||
Ramified primes: | \(256048319\), \(15776\!\cdots\!73317\) | 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: | $\Q(\sqrt{-40395\!\cdots\!04123}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $17$ | 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: | $a^{34}+2$, $a^{34}+a+1$, $a^{34}+a^{31}-a^{29}+a^{27}-a^{25}+a^{23}-a^{21}+a^{19}-a^{17}+a^{15}-a^{13}+a^{11}-a^{9}+a^{7}-a^{5}+a^{3}-a+2$, $a^{27}-a^{20}+a^{13}-a^{6}+1$, $a^{34}+a^{31}+a^{30}+a^{27}+a^{26}+a^{23}+a^{22}+a^{19}+a^{18}+a^{15}+a^{14}+a^{11}+a^{10}-a^{9}-a^{8}+a^{7}+a^{6}-a^{5}-a^{4}+a^{3}+a^{2}-a+1$, $a^{34}+a^{33}-a^{31}-a^{30}+a^{28}+a^{27}-a^{25}-a^{24}+a^{22}+a^{21}-a^{19}-a^{18}+a^{16}+a^{15}-a^{13}-a^{12}+a^{10}+a^{9}+a^{8}-a^{7}-a^{6}-a^{5}+a^{4}+a^{3}+a^{2}-a+1$, $a^{29}-a^{24}+a^{19}-a^{14}+a^{9}-a^{4}+1$, $a^{32}+2a^{31}+a^{30}-a^{28}-a^{26}-a^{25}-a^{24}+a^{23}+a^{22}+a^{21}-a^{18}-2a^{17}-a^{16}+a^{12}+2a^{11}+a^{10}-a^{9}-2a^{8}-a^{6}+2a^{3}+a^{2}+a$, $a^{33}-a^{32}+2a^{31}-a^{30}-a^{29}+a^{27}+a^{26}-2a^{25}+a^{24}-a^{18}+2a^{17}+a^{16}-2a^{15}+2a^{12}-2a^{11}+a^{10}+a^{9}-2a^{8}+a^{7}-a^{6}+3a^{5}-4a^{4}+2a^{3}+a^{2}-a+1$, $a^{34}+4a^{33}-4a^{31}+3a^{29}+a^{28}-2a^{27}-2a^{26}+a^{25}+3a^{24}-3a^{22}-a^{21}+3a^{20}+3a^{19}-2a^{18}-4a^{17}+2a^{16}+6a^{15}-a^{14}-5a^{13}+a^{12}+5a^{11}+a^{10}-2a^{9}-3a^{8}+2a^{7}+5a^{6}-5a^{4}-a^{3}+4a^{2}+5a-2$, $a^{34}+a^{33}-a^{32}-2a^{31}-3a^{30}-2a^{29}-a^{28}+a^{26}+a^{25}+2a^{24}-a^{22}-2a^{21}-a^{20}+a^{18}+2a^{17}+3a^{16}+3a^{15}+2a^{14}-a^{13}-a^{12}-3a^{11}-a^{10}-2a^{9}+a^{7}+2a^{6}+a^{5}-a^{4}-2a^{3}-3a^{2}-2a+1$, $2a^{34}+4a^{33}+4a^{32}+5a^{31}+4a^{30}+4a^{29}+5a^{28}+3a^{27}+3a^{26}+2a^{25}-2a^{24}-2a^{23}-3a^{22}-5a^{21}-4a^{20}-6a^{19}-6a^{18}-4a^{17}-5a^{16}-a^{15}+a^{14}-a^{13}+3a^{12}+3a^{11}+3a^{10}+7a^{9}+5a^{8}+6a^{7}+5a^{6}+a^{5}+4a^{4}+a^{3}-2a^{2}-1$, $2a^{34}+a^{30}-a^{27}-a^{26}-a^{25}-a^{22}-a^{21}+a^{19}+a^{15}+a^{13}+a^{12}+a^{11}-a^{10}+a^{8}+a^{7}-2a^{6}-2a^{5}+2a^{3}-a^{2}-2a+2$, $a^{34}-3a^{33}-3a^{31}+2a^{29}+a^{28}-4a^{24}+2a^{23}-a^{22}+4a^{21}+a^{20}-2a^{18}+a^{17}-2a^{16}-2a^{15}+5a^{14}-a^{13}+2a^{12}-3a^{11}-4a^{9}+2a^{8}-3a^{7}+4a^{6}+4a^{5}-4a^{4}-2a^{2}+a-1$, $a^{34}+2a^{33}+2a^{32}-a^{31}+a^{30}+2a^{29}-3a^{28}-3a^{27}-3a^{26}+3a^{21}+a^{20}-a^{19}+4a^{18}+3a^{17}-a^{16}-3a^{15}-4a^{14}-2a^{12}-a^{11}+3a^{9}-2a^{7}+5a^{6}+4a^{5}+a^{4}-3a^{3}-a^{2}-a$, $a^{34}+a^{33}-a^{31}+a^{29}+a^{25}-a^{23}+a^{20}-a^{19}-a^{12}-a^{11}+a^{10}-a^{7}+2a^{6}-a^{4}-a^{3}+2a^{2}+a$, $4a^{34}-a^{32}-5a^{31}-3a^{30}-3a^{29}-4a^{28}+2a^{27}+a^{26}+6a^{25}+5a^{24}-a^{23}-5a^{21}-2a^{20}-a^{18}+4a^{17}+2a^{16}+6a^{15}+4a^{14}-3a^{13}-2a^{12}-9a^{11}-3a^{10}-2a^{8}+5a^{7}+a^{6}+3a^{5}+5a^{4}-5a^{3}-2a^{2}-8a+6$ (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: | \( 363349966210764900 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{17}\cdot 363349966210764900 \cdot 1}{2\cdot\sqrt{403955037978023056104591172895949491439236366378374168898704123}}\cr\approx \mathstrut & 0.670218438782014 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 10333147966386144929666651337523200000000 |
The 14883 conjugacy class representatives for $S_{35}$ are not computed |
Character table for $S_{35}$ 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 | ${\href{/padicField/2.12.0.1}{12} }^{2}{,}\,{\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.3.0.1}{3} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | $32{,}\,{\href{/padicField/3.3.0.1}{3} }$ | $24{,}\,{\href{/padicField/5.11.0.1}{11} }$ | $18{,}\,{\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/11.13.0.1}{13} }$ | $17{,}\,16{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.5.0.1}{5} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | $15{,}\,{\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}$ | $34{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | $21{,}\,{\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | $29{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $35$ | $16{,}\,{\href{/padicField/59.9.0.1}{9} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(256048319\) | $\Q_{256048319}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $32$ | $1$ | $32$ | $0$ | 32T33 | $[\ ]^{32}$ | ||
\(157\!\cdots\!317\) | $\Q_{15\!\cdots\!17}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ |