Normalized defining polynomial
\( x^{4} + 77350x^{2} + 692723395 \)
Invariants
Degree: | $4$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(39697708482432000\) \(\medspace = 2^{10}\cdot 3^{2}\cdot 5^{3}\cdot 7^{3}\cdot 11^{2}\cdot 13^{2}\cdot 17^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(14\,115.34\) | 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: | $2^{11/4}3^{1/2}5^{3/4}7^{3/4}11^{1/2}13^{1/2}17^{3/4}\approx 16786.06337600352$ | ||
Ramified primes: | \(2\), \(3\), \(5\), \(7\), \(11\), \(13\), \(17\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{595}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | 4.0.601480431552000.9$^{2}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{143}a^{2}+\frac{5}{11}$, $\frac{1}{11869}a^{3}-\frac{358}{913}a$
Monogenic: | No | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{4}\times C_{4}\times C_{4}\times C_{4}\times C_{2856}$, which has order $1462272$ (assuming GRH)
Unit group
Rank: | $1$ | 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 unit: | $\frac{6}{143}a^{2}+\frac{4771}{11}$ (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: | \( 15.5480298008 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{2}\cdot 15.5480298008 \cdot 1462272}{2\cdot\sqrt{39697708482432000}}\cr\approx \mathstrut & 2.25242609645 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 8 |
The 5 conjugacy class representatives for $D_{4}$ |
Character table for $D_{4}$ |
Intermediate fields
\(\Q(\sqrt{39270}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 8 |
Degree 4 sibling: | 4.0.601480431552000.9 |
Minimal sibling: | 4.0.601480431552000.9 |
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 | R | R | R | R | R | ${\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }$ | ${\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.10.6 | $x^{4} + 4 x^{3} + 12 x^{2} + 10$ | $4$ | $1$ | $10$ | $D_{4}$ | $[2, 3, 7/2]$ |
\(3\) | 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(5\) | 5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
\(7\) | 7.4.3.1 | $x^{4} + 7$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |
\(11\) | 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(13\) | 13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(17\) | 17.4.3.1 | $x^{4} + 17$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |