Normalized defining polynomial
\( x^{10} - 15x^{8} - 4x^{7} + 21x^{6} + 238x^{5} + 39x^{4} - 172x^{3} - 262x^{2} - 782x - 972 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(294109629917470367744\) \(\medspace = 2^{13}\cdot 47^{3}\cdot 7019^{3}\) | 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: | \(111.39\) | 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: | $2^{11/6}47^{1/2}7019^{1/2}\approx 2046.797489224462$ | ||
Ramified primes: | \(2\), \(47\), \(7019\) | 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: | $\Q(\sqrt{659786}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{12}a^{7}-\frac{1}{4}a^{6}+\frac{5}{12}a^{5}+\frac{1}{4}a^{4}-\frac{1}{6}a^{3}+\frac{1}{6}a$, $\frac{1}{12}a^{8}+\frac{1}{6}a^{6}+\frac{1}{12}a^{4}+\frac{1}{6}a^{2}-\frac{1}{2}a$, $\frac{1}{213700308}a^{9}+\frac{2309425}{106850154}a^{8}+\frac{2341409}{213700308}a^{7}-\frac{25485515}{213700308}a^{6}+\frac{34854059}{106850154}a^{5}+\frac{3285005}{213700308}a^{4}-\frac{4016189}{106850154}a^{3}-\frac{8368591}{106850154}a^{2}+\frac{8662015}{35616718}a-\frac{1108322}{17808359}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$, $3$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $5$ | 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: | $\frac{145071929}{213700308}a^{9}+\frac{345658049}{213700308}a^{8}-\frac{570373841}{71233436}a^{7}-\frac{5111719001}{213700308}a^{6}-\frac{652440859}{35616718}a^{5}+\frac{8266639772}{53425077}a^{4}+\frac{13217017287}{35616718}a^{3}+\frac{17967716377}{53425077}a^{2}+\frac{8022497008}{53425077}a+\frac{1220420369}{17808359}$, $\frac{18618015}{17808359}a^{9}+\frac{79959867}{35616718}a^{8}-\frac{870957447}{71233436}a^{7}-\frac{2115516471}{71233436}a^{6}-\frac{2403674451}{71233436}a^{5}+\frac{15479746901}{71233436}a^{4}+\frac{15774793635}{35616718}a^{3}+\frac{10934325879}{17808359}a^{2}+\frac{29723257307}{35616718}a+\frac{11085015916}{17808359}$, $\frac{243949461}{71233436}a^{9}+\frac{89195554}{17808359}a^{8}-\frac{4117041611}{71233436}a^{7}-\frac{4595211173}{71233436}a^{6}+\frac{2437977006}{17808359}a^{5}+\frac{45132165397}{71233436}a^{4}+\frac{22943307278}{17808359}a^{3}-\frac{72331343711}{35616718}a^{2}+\frac{113844648005}{35616718}a-\frac{88966956086}{17808359}$, $\frac{6926618872157}{213700308}a^{9}+\frac{6232910814635}{106850154}a^{8}-\frac{6696782040966}{17808359}a^{7}-\frac{87287990499473}{106850154}a^{6}-\frac{57264073946067}{71233436}a^{5}+\frac{331006264874263}{53425077}a^{4}+\frac{447055687729131}{35616718}a^{3}+\frac{18\!\cdots\!67}{106850154}a^{2}+\frac{12\!\cdots\!35}{53425077}a+\frac{305302560355777}{17808359}$, $\frac{52\!\cdots\!13}{53425077}a^{9}+\frac{19\!\cdots\!15}{53425077}a^{8}-\frac{19\!\cdots\!33}{71233436}a^{7}-\frac{10\!\cdots\!51}{213700308}a^{6}+\frac{18\!\cdots\!59}{71233436}a^{5}+\frac{12\!\cdots\!91}{213700308}a^{4}-\frac{43\!\cdots\!65}{35616718}a^{3}+\frac{30\!\cdots\!15}{53425077}a^{2}-\frac{62\!\cdots\!21}{106850154}a-\frac{18\!\cdots\!68}{17808359}$ (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: | \( 64714019.533 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 64714019.533 \cdot 1}{2\cdot\sqrt{294109629917470367744}}\cr\approx \mathstrut & 11.762326283 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 120 |
The 7 conjugacy class representatives for $S_5$ |
Character table for $S_5$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 5 sibling: | 5.1.10556576.1 |
Degree 6 sibling: | 6.2.73527407479367591936.1 |
Degree 10 sibling: | data not computed |
Degree 12 sibling: | data not computed |
Degree 15 sibling: | data not computed |
Degree 20 siblings: | data not computed |
Degree 24 sibling: | data not computed |
Degree 30 siblings: | data not computed |
Degree 40 sibling: | data not computed |
Minimal sibling: | 5.1.10556576.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.2.0.1}{2} }^{3}{,}\,{\href{/padicField/3.1.0.1}{1} }^{4}$ | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | R | ${\href{/padicField/53.3.0.1}{3} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.3.0.1}{3} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 | $[\ ]$ |
2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
2.6.11.1 | $x^{6} + 4 x^{3} + 2$ | $6$ | $1$ | $11$ | $D_{6}$ | $[3]_{3}^{2}$ | |
\(47\) | $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(7019\) | $\Q_{7019}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{7019}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ |