Normalized defining polynomial
\( x^{10} - 3 x^{9} + 2347 x^{8} - 6384 x^{7} + 1766089 x^{6} - 4424351 x^{5} + 527063913 x^{4} + \cdots + 1752294565441 \)
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: | \(-86537167397365512024785664\) \(\medspace = -\,2^{8}\cdot 3^{5}\cdot 631\cdot 4663^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(392.39\) | 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))
| |
Galois root discriminant: | $2^{4/5}3^{1/2}631^{1/2}4663^{1/2}\approx 5172.875324201175$ | ||
Ramified primes: | \(2\), \(3\), \(631\), \(4663\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-8827059}$) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | deg 32$^{16}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{52\!\cdots\!63}a^{9}-\frac{86\!\cdots\!82}{52\!\cdots\!63}a^{8}+\frac{15\!\cdots\!35}{52\!\cdots\!63}a^{7}-\frac{12\!\cdots\!49}{52\!\cdots\!63}a^{6}+\frac{57\!\cdots\!04}{52\!\cdots\!63}a^{5}+\frac{25\!\cdots\!78}{52\!\cdots\!63}a^{4}+\frac{25\!\cdots\!89}{52\!\cdots\!63}a^{3}+\frac{22\!\cdots\!06}{52\!\cdots\!63}a^{2}+\frac{22\!\cdots\!33}{52\!\cdots\!63}a+\frac{14\!\cdots\!66}{52\!\cdots\!63}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{1018200}$, which has order $2036400$ (assuming GRH)
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: | $\frac{10\!\cdots\!98}{52\!\cdots\!63}a^{9}-\frac{23\!\cdots\!04}{52\!\cdots\!63}a^{8}+\frac{22\!\cdots\!76}{52\!\cdots\!63}a^{7}-\frac{51\!\cdots\!58}{52\!\cdots\!63}a^{6}+\frac{15\!\cdots\!30}{52\!\cdots\!63}a^{5}-\frac{32\!\cdots\!19}{52\!\cdots\!63}a^{4}+\frac{37\!\cdots\!71}{52\!\cdots\!63}a^{3}-\frac{71\!\cdots\!85}{52\!\cdots\!63}a^{2}+\frac{26\!\cdots\!62}{52\!\cdots\!63}a-\frac{29\!\cdots\!05}{52\!\cdots\!63}$, $\frac{26\!\cdots\!10}{52\!\cdots\!63}a^{9}-\frac{89\!\cdots\!38}{52\!\cdots\!63}a^{8}+\frac{84\!\cdots\!64}{52\!\cdots\!63}a^{7}-\frac{24\!\cdots\!57}{52\!\cdots\!63}a^{6}+\frac{86\!\cdots\!75}{52\!\cdots\!63}a^{5}-\frac{20\!\cdots\!13}{52\!\cdots\!63}a^{4}+\frac{29\!\cdots\!70}{52\!\cdots\!63}a^{3}-\frac{60\!\cdots\!69}{52\!\cdots\!63}a^{2}+\frac{23\!\cdots\!31}{52\!\cdots\!63}a-\frac{44\!\cdots\!24}{52\!\cdots\!63}$, $\frac{12\!\cdots\!08}{52\!\cdots\!63}a^{9}-\frac{32\!\cdots\!42}{52\!\cdots\!63}a^{8}+\frac{30\!\cdots\!40}{52\!\cdots\!63}a^{7}-\frac{75\!\cdots\!15}{52\!\cdots\!63}a^{6}+\frac{23\!\cdots\!05}{52\!\cdots\!63}a^{5}-\frac{53\!\cdots\!32}{52\!\cdots\!63}a^{4}+\frac{67\!\cdots\!41}{52\!\cdots\!63}a^{3}-\frac{13\!\cdots\!54}{52\!\cdots\!63}a^{2}+\frac{50\!\cdots\!93}{52\!\cdots\!63}a-\frac{68\!\cdots\!66}{52\!\cdots\!63}$, $\frac{48\!\cdots\!89}{52\!\cdots\!63}a^{9}-\frac{72\!\cdots\!84}{52\!\cdots\!63}a^{8}+\frac{11\!\cdots\!44}{52\!\cdots\!63}a^{7}-\frac{18\!\cdots\!65}{52\!\cdots\!63}a^{6}+\frac{91\!\cdots\!83}{52\!\cdots\!63}a^{5}-\frac{15\!\cdots\!07}{52\!\cdots\!63}a^{4}+\frac{26\!\cdots\!23}{52\!\cdots\!63}a^{3}-\frac{47\!\cdots\!35}{52\!\cdots\!63}a^{2}+\frac{19\!\cdots\!27}{52\!\cdots\!63}a-\frac{40\!\cdots\!52}{52\!\cdots\!63}$ (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: | \( 285.9493115399786 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 285.9493115399786 \cdot 2036400}{2\cdot\sqrt{86537167397365512024785664}}\cr\approx \mathstrut & 0.306492712018128 \end{aligned}\] (assuming GRH)
Galois group
$C_2\wr S_5$ (as 10T39):
A non-solvable group of order 3840 |
The 36 conjugacy class representatives for $C_2 \wr S_5$ |
Character table for $C_2 \wr S_5$ |
Intermediate fields
5.5.223824.1 |
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 20 siblings: | data not computed |
Degree 30 siblings: | data not computed |
Degree 32 siblings: | data not computed |
Degree 40 siblings: | data not computed |
Minimal sibling: | 10.0.31611322457856.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ | ${\href{/padicField/13.10.0.1}{10} }$ | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.5.0.1}{5} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 | $x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 55 x^{5} + 55 x^{4} + 10 x^{3} - 25 x^{2} - 5 x + 7$ | $5$ | $2$ | $8$ | $F_5$ | $[\ ]_{5}^{4}$ |
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(631\) | $\Q_{631}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{631}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
\(4663\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $2$ | $3$ | $3$ |