Normalized defining polynomial
\( x^{7} - x^{6} - 735x^{4} - 210x^{3} - 88200x + 7200 \)
Invariants
Degree: | $7$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[3, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(1267156168933150486130625\) \(\medspace = 3^{10}\cdot 5^{4}\cdot 47^{2}\cdot 3942493^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(2774.99\) | 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: | $3^{11/6}5^{2/3}47^{1/2}3942493^{1/2}\approx 298288.62872654863$ | ||
Ramified primes: | \(3\), \(5\), \(47\), \(3942493\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{270}a^{4}-\frac{1}{270}a^{3}+\frac{7}{18}a+\frac{1}{9}$, $\frac{1}{540}a^{5}-\frac{1}{540}a^{4}+\frac{7}{36}a^{2}+\frac{1}{18}a$, $\frac{1}{1080}a^{6}-\frac{1}{1080}a^{5}+\frac{7}{72}a^{3}+\frac{1}{36}a^{2}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (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: | $\frac{17\!\cdots\!27}{540}a^{6}-\frac{33\!\cdots\!09}{180}a^{5}+\frac{24\!\cdots\!57}{270}a^{4}-\frac{16\!\cdots\!31}{60}a^{3}+\frac{76\!\cdots\!91}{6}a^{2}-\frac{10\!\cdots\!77}{18}a+\frac{14\!\cdots\!27}{3}$, $\frac{47\!\cdots\!23}{540}a^{6}-\frac{43\!\cdots\!93}{540}a^{5}-\frac{29\!\cdots\!88}{45}a^{4}-\frac{11\!\cdots\!93}{180}a^{3}-\frac{21\!\cdots\!61}{9}a^{2}-\frac{11\!\cdots\!13}{6}a-\frac{23\!\cdots\!55}{3}$, $\frac{19\!\cdots\!25}{216}a^{6}-\frac{12\!\cdots\!53}{1080}a^{5}-\frac{92\!\cdots\!83}{54}a^{4}-\frac{80\!\cdots\!87}{1080}a^{3}-\frac{26\!\cdots\!19}{36}a^{2}+\frac{24\!\cdots\!57}{18}a-\frac{99\!\cdots\!33}{9}$, $\frac{16\!\cdots\!73}{1080}a^{6}-\frac{84\!\cdots\!43}{216}a^{5}-\frac{66\!\cdots\!61}{60}a^{4}+\frac{32\!\cdots\!53}{216}a^{3}-\frac{40\!\cdots\!68}{9}a^{2}-\frac{23\!\cdots\!57}{2}a+\frac{84\!\cdots\!77}{9}$ (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: | \( 26717545658.4 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{2}\cdot 26717545658.4 \cdot 1}{2\cdot\sqrt{1267156168933150486130625}}\cr\approx \mathstrut & 3.74801410991 \end{aligned}\] (assuming GRH)
Galois group
$\GL(3,2)$ (as 7T5):
A non-solvable group of order 168 |
The 6 conjugacy class representatives for $\GL(3,2)$ |
Character table for $\GL(3,2)$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 8 sibling: | deg 8 |
Degree 14 siblings: | deg 14, deg 14 |
Degree 21 sibling: | deg 21 |
Degree 24 sibling: | deg 24 |
Degree 28 sibling: | deg 28 |
Degree 42 siblings: | deg 42, deg 42, deg 42 |
Arithmetically equvalently sibling: | 7.3.1267156168933150486130625.2 |
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 | ${\href{/padicField/2.3.0.1}{3} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | R | R | ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.7.0.1}{7} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | R | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.7.0.1}{7} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.3.5.1 | $x^{3} + 3$ | $3$ | $1$ | $5$ | $S_3$ | $[5/2]_{2}$ | |
3.3.5.1 | $x^{3} + 3$ | $3$ | $1$ | $5$ | $S_3$ | $[5/2]_{2}$ | |
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
5.3.2.1 | $x^{3} + 5$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
5.3.2.1 | $x^{3} + 5$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(47\) | $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
47.2.1.1 | $x^{2} + 235$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(3942493\) | $\Q_{3942493}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
3.625...225.42t37.a.a | $3$ | $ 3^{6} \cdot 5^{2} \cdot 47^{2} \cdot 3942493^{2}$ | 7.3.1267156168933150486130625.1 | $\GL(3,2)$ (as 7T5) | $0$ | $-1$ | |
3.625...225.42t37.a.b | $3$ | $ 3^{6} \cdot 5^{2} \cdot 47^{2} \cdot 3942493^{2}$ | 7.3.1267156168933150486130625.1 | $\GL(3,2)$ (as 7T5) | $0$ | $-1$ | |
* | 6.126...625.7t5.a.a | $6$ | $ 3^{10} \cdot 5^{4} \cdot 47^{2} \cdot 3942493^{2}$ | 7.3.1267156168933150486130625.1 | $\GL(3,2)$ (as 7T5) | $1$ | $2$ |
7.391...625.8t37.a.a | $7$ | $ 3^{12} \cdot 5^{4} \cdot 47^{4} \cdot 3942493^{4}$ | 7.3.1267156168933150486130625.1 | $\GL(3,2)$ (as 7T5) | $1$ | $-1$ | |
8.792...625.21t14.a.a | $8$ | $ 3^{16} \cdot 5^{6} \cdot 47^{4} \cdot 3942493^{4}$ | 7.3.1267156168933150486130625.1 | $\GL(3,2)$ (as 7T5) | $1$ | $0$ |