Normalized defining polynomial
\( x^{35} - 4x - 4 \)
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: | \(-18133024539532207970634996343261673883948543963577136855073685504\) \(\medspace = -\,2^{34}\cdot 2393\cdot 3027347\cdot 3480929\cdot 23465153\cdot 86816318477729\cdot 20545919111781857\) | 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: | \(68.54\) | 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: | $2^{34/35}2393^{1/2}3027347^{1/2}3480929^{1/2}23465153^{1/2}86816318477729^{1/2}20545919111781857^{1/2}\approx 2.014440002426347e+27$ | ||
Ramified primes: | \(2\), \(2393\), \(3027347\), \(3480929\), \(23465153\), \(86816318477729\), \(20545919111781857\) | 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: | $\Q(\sqrt{-10554\!\cdots\!68731}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2}a^{18}$, $\frac{1}{2}a^{19}$, $\frac{1}{2}a^{20}$, $\frac{1}{2}a^{21}$, $\frac{1}{2}a^{22}$, $\frac{1}{2}a^{23}$, $\frac{1}{2}a^{24}$, $\frac{1}{2}a^{25}$, $\frac{1}{2}a^{26}$, $\frac{1}{2}a^{27}$, $\frac{1}{2}a^{28}$, $\frac{1}{2}a^{29}$, $\frac{1}{2}a^{30}$, $\frac{1}{2}a^{31}$, $\frac{1}{2}a^{32}$, $\frac{1}{2}a^{33}$, $\frac{1}{2}a^{34}$
Monogenic: | Not computed | |
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)
| |
Fundamental units: | $\frac{1}{2}a^{18}-a-1$, $\frac{1}{2}a^{18}+a+1$, $\frac{1}{2}a^{18}-a^{9}+1$, $\frac{1}{2}a^{27}-\frac{1}{2}a^{26}+\frac{1}{2}a^{25}-\frac{1}{2}a^{24}+\frac{1}{2}a^{23}-\frac{1}{2}a^{22}+\frac{1}{2}a^{21}+1$, $a^{34}-\frac{1}{2}a^{33}+\frac{1}{2}a^{32}-\frac{1}{2}a^{31}+\frac{1}{2}a^{30}-\frac{1}{2}a^{28}+\frac{1}{2}a^{27}-\frac{1}{2}a^{26}+\frac{1}{2}a^{25}-a^{24}+a^{23}-a^{22}+\frac{1}{2}a^{21}-a^{20}+\frac{1}{2}a^{19}-a^{18}+a^{12}-a^{11}+a^{10}+a^{8}-a^{7}+2a^{6}+a^{2}-a-5$, $a^{12}-a-1$, $\frac{3}{2}a^{34}-\frac{1}{2}a^{33}-\frac{1}{2}a^{32}+a^{31}-\frac{3}{2}a^{30}+\frac{3}{2}a^{29}-a^{28}+\frac{1}{2}a^{27}+\frac{1}{2}a^{26}-a^{25}+\frac{3}{2}a^{24}-2a^{23}+2a^{22}-\frac{3}{2}a^{21}+a^{20}-a^{18}+2a^{17}-3a^{16}+3a^{15}-3a^{14}+3a^{13}-2a^{12}+a^{11}-a^{9}+2a^{8}-2a^{7}+2a^{6}-a^{5}+a^{3}-2a^{2}+3a-9$, $a^{33}-a^{32}+\frac{1}{2}a^{29}-a^{28}+\frac{1}{2}a^{26}-a^{25}+\frac{1}{2}a^{24}-\frac{1}{2}a^{23}-\frac{1}{2}a^{21}+a^{20}-\frac{1}{2}a^{19}-a^{18}+2a^{17}-a^{15}+3a^{13}-a^{12}-2a^{11}+3a^{10}+a^{9}-2a^{8}-a^{7}+4a^{6}-a^{5}-3a^{4}+2a^{3}+2a^{2}-2a-3$, $\frac{1}{2}a^{34}+\frac{1}{2}a^{33}-\frac{1}{2}a^{30}-a^{29}+a^{28}-\frac{1}{2}a^{27}+\frac{1}{2}a^{26}-\frac{3}{2}a^{24}+a^{23}+\frac{1}{2}a^{22}+a^{21}+\frac{3}{2}a^{20}-\frac{1}{2}a^{19}+a^{17}+2a^{15}-a^{14}-2a^{13}-a^{12}-2a^{11}+a^{10}-3a^{8}-a^{7}-2a^{6}+4a^{4}+a^{2}-a-3$, $\frac{3}{2}a^{34}-\frac{1}{2}a^{32}-\frac{1}{2}a^{30}+\frac{3}{2}a^{29}-\frac{1}{2}a^{28}-\frac{1}{2}a^{27}-\frac{1}{2}a^{26}+\frac{3}{2}a^{25}-\frac{1}{2}a^{24}+\frac{1}{2}a^{23}-\frac{5}{2}a^{22}+\frac{5}{2}a^{21}-a^{20}+a^{19}-\frac{3}{2}a^{18}+2a^{16}-2a^{15}-a^{13}+3a^{12}-2a^{10}-2a^{9}+2a^{8}+a^{7}+a^{6}-4a^{5}+2a^{4}+a^{3}-a^{2}-a-7$, $9a^{34}-\frac{11}{2}a^{33}+\frac{1}{2}a^{32}+\frac{9}{2}a^{31}-9a^{30}+12a^{29}-13a^{28}+12a^{27}-9a^{26}+\frac{9}{2}a^{25}+a^{24}-\frac{13}{2}a^{23}+\frac{23}{2}a^{22}-14a^{21}+\frac{29}{2}a^{20}-12a^{19}+\frac{17}{2}a^{18}-3a^{17}-3a^{16}+9a^{15}-14a^{14}+16a^{13}-16a^{12}+12a^{11}-8a^{10}+a^{9}+5a^{8}-12a^{7}+17a^{6}-18a^{5}+17a^{4}-12a^{3}+6a^{2}+a-45$, $a^{34}+a^{32}-\frac{5}{2}a^{31}-a^{29}+2a^{28}-\frac{3}{2}a^{27}-\frac{1}{2}a^{26}-\frac{3}{2}a^{25}+\frac{3}{2}a^{24}+\frac{3}{2}a^{23}+\frac{1}{2}a^{22}-\frac{1}{2}a^{21}-a^{20}+3a^{19}+\frac{3}{2}a^{18}+a^{17}-4a^{16}+2a^{13}-3a^{12}-3a^{11}-3a^{10}+2a^{9}+3a^{8}-a^{7}-3a^{6}+7a^{4}+4a^{3}+a^{2}-4a-3$, $\frac{1}{2}a^{32}+\frac{1}{2}a^{31}-\frac{1}{2}a^{30}+\frac{1}{2}a^{29}+\frac{1}{2}a^{28}+a^{26}-a^{25}+a^{24}-a^{23}+a^{22}+a^{21}-\frac{1}{2}a^{20}+\frac{1}{2}a^{19}-\frac{1}{2}a^{18}+2a^{17}-a^{15}+a^{13}+a^{12}+a^{11}-a^{10}+2a^{9}-a^{8}+2a^{7}+a^{6}+a^{4}+5a^{2}+a-1$, $a^{34}-\frac{1}{2}a^{33}+\frac{1}{2}a^{32}-a^{31}+\frac{1}{2}a^{30}+\frac{3}{2}a^{29}-\frac{3}{2}a^{28}-a^{27}+2a^{26}-\frac{1}{2}a^{25}-\frac{1}{2}a^{23}+2a^{21}-3a^{20}+\frac{1}{2}a^{19}+3a^{18}-2a^{17}-a^{16}+a^{14}+2a^{13}-4a^{12}+a^{11}+3a^{10}-4a^{9}+2a^{7}+a^{6}-a^{5}-5a^{4}+4a^{3}+4a^{2}-6a-3$, $\frac{1}{2}a^{31}-a^{30}+a^{29}-\frac{1}{2}a^{28}-\frac{1}{2}a^{26}+a^{25}-a^{24}+\frac{1}{2}a^{23}-a^{22}+\frac{3}{2}a^{21}-\frac{1}{2}a^{20}-a^{19}+a^{18}-2a^{15}+a^{14}+a^{13}-2a^{11}+2a^{10}-a^{8}-2a^{7}+a^{6}+2a^{5}-a^{4}-a^{3}+a^{2}+a-3$, $a^{34}-a^{33}-\frac{1}{2}a^{32}+a^{31}-a^{29}+a^{28}+a^{27}-a^{26}+\frac{1}{2}a^{25}+a^{24}-a^{23}-\frac{1}{2}a^{22}-a^{20}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}+a^{16}+3a^{13}-2a^{11}+3a^{10}-5a^{8}+2a^{7}+2a^{6}-5a^{5}+5a^{3}-3a^{2}-3a+1$, $a^{34}-\frac{5}{2}a^{33}+\frac{5}{2}a^{32}-2a^{31}+a^{30}-\frac{1}{2}a^{29}+a^{28}-\frac{1}{2}a^{27}+\frac{1}{2}a^{24}-a^{23}-\frac{1}{2}a^{22}+2a^{21}-\frac{7}{2}a^{20}+2a^{19}-\frac{1}{2}a^{18}+a^{17}-a^{16}+a^{15}-2a^{12}+2a^{11}-a^{10}-3a^{9}+3a^{8}-2a^{7}+a^{6}+2a^{4}-a^{3}-a^{2}-1$ (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: | \( 2024409859615809000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{17}\cdot 2024409859615809000 \cdot 1}{2\cdot\sqrt{18133024539532207970634996343261673883948543963577136855073685504}}\cr\approx \mathstrut & 0.557341113922882 \end{aligned}\] (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 | R | $32{,}\,{\href{/padicField/3.3.0.1}{3} }$ | $26{,}\,{\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | $23{,}\,{\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/13.11.0.1}{11} }$ | $24{,}\,{\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $18{,}\,17$ | $18{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | $33{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $27{,}\,{\href{/padicField/31.8.0.1}{8} }$ | $17{,}\,15{,}\,{\href{/padicField/37.3.0.1}{3} }$ | $34{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $31{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.5.0.1}{5} }$ | $29{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 $35$ | $35$ | $1$ | $34$ | |||
\(2393\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
Deg $21$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ | ||
\(3027347\) | $\Q_{3027347}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
\(3480929\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $20$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ | ||
\(23465153\) | $\Q_{23465153}$ | $x$ | $1$ | $1$ | $0$ | 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 $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
\(86816318477729\) | $\Q_{86816318477729}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
\(20545919111781857\) | $\Q_{20545919111781857}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{20545919111781857}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $29$ | $1$ | $29$ | $0$ | $C_{29}$ | $[\ ]^{29}$ |