Normalized defining polynomial
\( x^{10} - x^{9} + 826 x^{8} - 826 x^{7} + 248326 x^{6} - 248326 x^{5} + 32732701 x^{4} + \cdots + 27876482701 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-5825948542184271384191\) \(\medspace = -\,7^{5}\cdot 11^{9}\cdot 43^{5}\) | 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: | \(150.15\) | 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: | $7^{1/2}11^{9/10}43^{1/2}\approx 150.15391648314187$ | ||
Ramified primes: | \(7\), \(11\), \(43\) | 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: | \(\Q(\sqrt{-3311}) \) | ||
$\card{ \Gal(K/\Q) }$: | $10$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(3311=7\cdot 11\cdot 43\) | ||
Dirichlet character group: | $\lbrace$$\chi_{3311}(1504,·)$, $\chi_{3311}(1,·)$, $\chi_{3311}(3009,·)$, $\chi_{3311}(302,·)$, $\chi_{3311}(1807,·)$, $\chi_{3311}(2708,·)$, $\chi_{3311}(3310,·)$, $\chi_{3311}(2710,·)$, $\chi_{3311}(601,·)$, $\chi_{3311}(603,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | \(\Q(\sqrt{-3311}) \), 10.0.5825948542184271384191.1$^{15}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2862580051}a^{6}-\frac{189827564}{2862580051}a^{5}+\frac{450}{2862580051}a^{4}+\frac{379164775}{2862580051}a^{3}+\frac{50625}{2862580051}a^{2}-\frac{188442385}{2862580051}a+\frac{843750}{2862580051}$, $\frac{1}{2862580051}a^{7}+\frac{525}{2862580051}a^{5}-\frac{75832955}{2862580051}a^{4}+\frac{78750}{2862580051}a^{3}+\frac{150753908}{2862580051}a^{2}+\frac{2953125}{2862580051}a-\frac{71888552}{2862580051}$, $\frac{1}{2862580051}a^{8}-\frac{606663640}{2862580051}a^{5}-\frac{157500}{2862580051}a^{4}-\frac{1392729448}{2862580051}a^{3}-\frac{23625000}{2862580051}a^{2}-\frac{1329938212}{2862580051}a-\frac{442968750}{2862580051}$, $\frac{1}{2862580051}a^{9}-\frac{202500}{2862580051}a^{5}-\frac{339196293}{2862580051}a^{4}-\frac{40500000}{2862580051}a^{3}+\frac{1258049660}{2862580051}a^{2}+\frac{1153986301}{2862580051}a+\frac{194430435}{2862580051}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{5436}$, which has order $173952$ (assuming GRH)
Unit group
Rank: | $4$ | 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: | $\frac{5851}{2862580051}a^{6}-\frac{17176}{2862580051}a^{5}+\frac{2632950}{2862580051}a^{4}-\frac{6441000}{2862580051}a^{3}+\frac{296206875}{2862580051}a^{2}-\frac{483075000}{2862580051}a+\frac{4936781250}{2862580051}$, $\frac{76}{2862580051}a^{8}+\frac{45600}{2862580051}a^{6}+\frac{8550000}{2862580051}a^{4}-\frac{1744201}{2862580051}a^{3}+\frac{513000000}{2862580051}a^{2}-\frac{392445225}{2862580051}a+\frac{4809375000}{2862580051}$, $\frac{1}{2862580051}a^{9}+\frac{826}{2862580051}a^{7}-\frac{5851}{2862580051}a^{6}+\frac{248326}{2862580051}a^{5}-\frac{3088951}{2862580051}a^{4}+\frac{30988500}{2862580051}a^{3}-\frac{468951451}{2862580051}a^{2}+\frac{1213762500}{2862580051}a-\frac{18321013951}{2862580051}$, $\frac{1}{2862580051}a^{9}-\frac{76}{2862580051}a^{8}+\frac{826}{2862580051}a^{7}-\frac{51451}{2862580051}a^{6}+\frac{248326}{2862580051}a^{5}-\frac{11638951}{2862580051}a^{4}+\frac{32732701}{2862580051}a^{3}-\frac{981951451}{2862580051}a^{2}+\frac{1606207725}{2862580051}a-\frac{23130388951}{2862580051}$ (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: | \( 26.1711060094 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 26.1711060094 \cdot 173952}{2\cdot\sqrt{5825948542184271384191}}\cr\approx \mathstrut & 0.292036763639 \end{aligned}\] (assuming GRH)
Galois group
A cyclic group of order 10 |
The 10 conjugacy class representatives for $C_{10}$ |
Character table for $C_{10}$ |
Intermediate fields
\(\Q(\sqrt{-3311}) \), \(\Q(\zeta_{11})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{2}$ | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | R | R | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }$ | ${\href{/padicField/23.1.0.1}{1} }^{10}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }$ | ${\href{/padicField/37.10.0.1}{10} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/47.10.0.1}{10} }$ | ${\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.10.5.2 | $x^{10} + 35 x^{8} + 492 x^{6} + 8 x^{5} + 3360 x^{4} - 560 x^{3} + 11516 x^{2} + 1968 x + 17516$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
\(11\) | 11.10.9.7 | $x^{10} + 11$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |
\(43\) | 43.2.1.1 | $x^{2} + 86$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
43.2.1.1 | $x^{2} + 86$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
43.2.1.1 | $x^{2} + 86$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
43.2.1.1 | $x^{2} + 86$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
43.2.1.1 | $x^{2} + 86$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |