Normalized defining polynomial
\( x^{10} - 3x^{9} + 4x^{7} + 36x^{6} - 48x^{5} + 101x^{4} - 61x^{3} + 420x^{2} - 90x + 736 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-11873367303361648\) \(\medspace = -\,2^{4}\cdot 7^{5}\cdot 210127^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(40.50\) | 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: | $2\cdot 7^{1/2}210127^{1/2}\approx 2425.6042546136828$ | ||
Ramified primes: | \(2\), \(7\), \(210127\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | unavailable$^{16}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{9726614566}a^{9}-\frac{2073773165}{9726614566}a^{8}-\frac{942773817}{9726614566}a^{7}-\frac{1566548865}{9726614566}a^{6}+\frac{1050958653}{4863307283}a^{5}-\frac{2900672507}{9726614566}a^{4}+\frac{258606994}{4863307283}a^{3}-\frac{1936999821}{9726614566}a^{2}-\frac{1406458806}{4863307283}a+\frac{999216170}{4863307283}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{130}$, which has order $130$
Unit group
Rank: | $4$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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: | $\frac{1631729}{9726614566}a^{9}+\frac{50810499}{4863307283}a^{8}-\frac{255504401}{9726614566}a^{7}-\frac{274850913}{9726614566}a^{6}+\frac{249001291}{4863307283}a^{5}+\frac{4628755079}{9726614566}a^{4}-\frac{1948688565}{4863307283}a^{3}+\frac{120503046}{4863307283}a^{2}-\frac{1042641145}{4863307283}a+\frac{17534533367}{4863307283}$, $\frac{25043050}{4863307283}a^{9}+\frac{109072470}{4863307283}a^{8}-\frac{72109231}{4863307283}a^{7}-\frac{284418528}{4863307283}a^{6}-\frac{431353745}{4863307283}a^{5}+\frac{2545356778}{4863307283}a^{4}-\frac{4122921878}{4863307283}a^{3}+\frac{872722887}{4863307283}a^{2}-\frac{1792909083}{4863307283}a+\frac{108238383}{4863307283}$, $\frac{20477713}{9726614566}a^{9}-\frac{6641895}{4863307283}a^{8}+\frac{180344553}{9726614566}a^{7}+\frac{284435711}{9726614566}a^{6}-\frac{699586914}{4863307283}a^{5}-\frac{2409488231}{9726614566}a^{4}-\frac{185929607}{4863307283}a^{3}+\frac{1273178668}{4863307283}a^{2}-\frac{371365584}{4863307283}a-\frac{1740552745}{4863307283}$, $\frac{48454371}{9726614566}a^{9}-\frac{58261971}{4863307283}a^{8}-\frac{111285939}{9726614566}a^{7}+\frac{293986143}{9726614566}a^{6}+\frac{680355036}{4863307283}a^{5}-\frac{461958477}{9726614566}a^{4}+\frac{2174233313}{4863307283}a^{3}-\frac{752219841}{4863307283}a^{2}+\frac{750267938}{4863307283}a+\frac{12562987701}{4863307283}$ | 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: | \( 904.860574966527 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 904.860574966527 \cdot 130}{2\cdot\sqrt{11873367303361648}}\cr\approx \mathstrut & 5.28575967331531 \end{aligned}\]
Galois group
$C_2\times S_5$ (as 10T22):
A non-solvable group of order 240 |
The 14 conjugacy class representatives for $S_5\times C_2$ |
Character table for $S_5\times C_2$ |
Intermediate fields
\(\Q(\sqrt{-7}) \), 5.5.840508.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 10 sibling: | data not computed |
Degree 12 siblings: | data not computed |
Degree 20 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 30 siblings: | data not computed |
Degree 40 siblings: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.10.0.1}{10} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{5}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{5}$ |
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 | $[\ ]$ |
$\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
\(7\) | 7.10.5.2 | $x^{10} + 35 x^{8} + 492 x^{6} + 8 x^{5} + 3360 x^{4} - 560 x^{3} + 11516 x^{2} + 1968 x + 17516$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
\(210127\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |