Normalized defining polynomial
\( x^{10} - 3x^{9} - x^{8} + 14x^{7} - 15x^{6} - 15x^{5} + 33x^{4} - 12x^{3} - 11x^{2} + 7x + 1 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-234367644416\) \(\medspace = -\,2^{8}\cdot 971^{3}\) | 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: | \(13.71\) | 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: | $2^{4/5}971^{1/2}\approx 54.254230914862234$ | ||
Ramified primes: | \(2\), \(971\) | 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: | \(\Q(\sqrt{-971}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{1513}a^{9}+\frac{450}{1513}a^{8}-\frac{406}{1513}a^{7}+\frac{682}{1513}a^{6}+\frac{279}{1513}a^{5}-\frac{720}{1513}a^{4}+\frac{681}{1513}a^{3}-\frac{171}{1513}a^{2}-\frac{311}{1513}a-\frac{167}{1513}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | 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: | $\frac{471}{1513}a^{9}-\frac{1383}{1513}a^{8}-\frac{588}{1513}a^{7}+\frac{6518}{1513}a^{6}-\frac{6274}{1513}a^{5}-\frac{7773}{1513}a^{4}+\frac{13612}{1513}a^{3}-\frac{3378}{1513}a^{2}-\frac{4259}{1513}a+\frac{1532}{1513}$, $\frac{462}{1513}a^{9}-\frac{894}{1513}a^{8}-\frac{1473}{1513}a^{7}+\frac{4919}{1513}a^{6}-\frac{1220}{1513}a^{5}-\frac{8858}{1513}a^{4}+\frac{4457}{1513}a^{3}+\frac{2700}{1513}a^{2}-\frac{1460}{1513}a-\frac{3017}{1513}$, $\frac{27}{1513}a^{9}+\frac{46}{1513}a^{8}-\frac{371}{1513}a^{7}+\frac{258}{1513}a^{6}+\frac{1481}{1513}a^{5}-\frac{2797}{1513}a^{4}-\frac{1282}{1513}a^{3}+\frac{4461}{1513}a^{2}-\frac{2345}{1513}a-\frac{1483}{1513}$, $\frac{571}{1513}a^{9}-\frac{1773}{1513}a^{8}-\frac{337}{1513}a^{7}+\frac{8146}{1513}a^{6}-\frac{10147}{1513}a^{5}-\frac{7149}{1513}a^{4}+\frac{21192}{1513}a^{3}-\frac{11400}{1513}a^{2}-\frac{6612}{1513}a+\frac{6014}{1513}$, $\frac{30}{1513}a^{9}-\frac{117}{1513}a^{8}-\frac{76}{1513}a^{7}+\frac{791}{1513}a^{6}-\frac{708}{1513}a^{5}-\frac{1931}{1513}a^{4}+\frac{3787}{1513}a^{3}+\frac{922}{1513}a^{2}-\frac{4791}{1513}a+\frac{2555}{1513}$, $\frac{866}{1513}a^{9}-\frac{2167}{1513}a^{8}-\frac{2093}{1513}a^{7}+\frac{11133}{1513}a^{6}-\frac{6518}{1513}a^{5}-\frac{16807}{1513}a^{4}+\frac{17832}{1513}a^{3}+\frac{1701}{1513}a^{2}-\frac{4551}{1513}a+\frac{626}{1513}$ | 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: | \( 72.2708766366 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 72.2708766366 \cdot 1}{2\cdot\sqrt{234367644416}}\cr\approx \mathstrut & 0.296240264794 \end{aligned}\]
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: | 5.3.15536.1 |
Degree 6 sibling: | 6.0.14647977776.2 |
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: | 5.3.15536.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.3.0.1}{3} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.10.8.1 | $x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 55 x^{5} + 55 x^{4} + 10 x^{3} - 25 x^{2} - 5 x + 7$ | $5$ | $2$ | $8$ | $F_5$ | $[\ ]_{5}^{4}$ |
\(971\) | $\Q_{971}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $6$ | $2$ | $3$ | $3$ |