Normalized defining polynomial
\( x^{10} - 3x^{9} - 2x^{8} + 10x^{7} + 41x^{6} - 71x^{5} + 90x^{4} - 140x^{3} + 602x^{2} - 16x + 1072 \)
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: | \(-10420954392927232\) \(\medspace = -\,2^{10}\cdot 7^{5}\cdot 11^{2}\cdot 2237^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(39.98\) | 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: | $2^{11/6}7^{1/2}11^{1/2}2237^{1/2}\approx 1478.99491223052$ | ||
Ramified primes: | \(2\), \(7\), \(11\), \(2237\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{35285400564}a^{9}+\frac{40131286}{8821350141}a^{8}-\frac{2921141723}{17642700282}a^{7}-\frac{162928029}{2940450047}a^{6}-\frac{13802404507}{35285400564}a^{5}-\frac{739106281}{2940450047}a^{4}+\frac{1564039011}{5880900094}a^{3}-\frac{1470125089}{17642700282}a^{2}-\frac{917638055}{2940450047}a-\frac{2580059752}{8821350141}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{104}$, which has order $104$
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: | $\frac{7249313}{8821350141}a^{9}-\frac{240047605}{35285400564}a^{8}+\frac{183726425}{17642700282}a^{7}+\frac{82489297}{2940450047}a^{6}-\frac{252315277}{17642700282}a^{5}-\frac{3654434165}{11761800188}a^{4}+\frac{1118740244}{2940450047}a^{3}+\frac{1482583699}{17642700282}a^{2}+\frac{3619933493}{5880900094}a-\frac{22443218519}{8821350141}$, $\frac{6906049}{8821350141}a^{9}-\frac{2455357}{35285400564}a^{8}+\frac{128767967}{17642700282}a^{7}+\frac{49299839}{2940450047}a^{6}-\frac{1371906757}{17642700282}a^{5}-\frac{1144440621}{11761800188}a^{4}+\frac{133090715}{2940450047}a^{3}+\frac{6353279065}{17642700282}a^{2}+\frac{1672252971}{5880900094}a+\frac{4280800315}{8821350141}$, $\frac{42900490}{8821350141}a^{9}-\frac{186404615}{8821350141}a^{8}+\frac{41277344}{8821350141}a^{7}+\frac{280054670}{2940450047}a^{6}+\frac{714665246}{8821350141}a^{5}-\frac{2057733346}{2940450047}a^{4}+\frac{2964947694}{2940450047}a^{3}-\frac{2814751289}{8821350141}a^{2}+\frac{3499680358}{2940450047}a-\frac{2602678009}{8821350141}$, $\frac{1157133}{2940450047}a^{9}+\frac{92517399}{11761800188}a^{8}-\frac{97221897}{5880900094}a^{7}-\frac{73891046}{2940450047}a^{6}+\frac{218280645}{5880900094}a^{5}+\frac{4735095353}{11761800188}a^{4}-\frac{1296226196}{2940450047}a^{3}-\frac{899373775}{5880900094}a^{2}-\frac{4656471883}{5880900094}a+\frac{24596515925}{2940450047}$ | 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: | \( 781.448847418073 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 781.448847418073 \cdot 104}{2\cdot\sqrt{10420954392927232}}\cr\approx \mathstrut & 3.89806873472646 \end{aligned}\]
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
\(\Q(\sqrt{-7}) \), 5.5.787424.1 |
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 | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.10.0.1}{10} }$ | R | R | ${\href{/padicField/13.10.0.1}{10} }$ | ${\href{/padicField/17.10.0.1}{10} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{5}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(7\) | 7.10.5.2 | $x^{10} + 35 x^{8} + 492 x^{6} + 8 x^{5} + 3360 x^{4} - 560 x^{3} + 11516 x^{2} + 1968 x + 17516$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
\(11\) | 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
\(2237\) | $\Q_{2237}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2237}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |