Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 15*x^18 + 86*x^16 - 222*x^14 + 158*x^12 + 468*x^10 - 1211*x^8 + 1137*x^6 - 474*x^4 + 72*x^2 - 1)
gp: K = bnfinit(y^20 - 15*y^18 + 86*y^16 - 222*y^14 + 158*y^12 + 468*y^10 - 1211*y^8 + 1137*y^6 - 474*y^4 + 72*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 15*x^18 + 86*x^16 - 222*x^14 + 158*x^12 + 468*x^10 - 1211*x^8 + 1137*x^6 - 474*x^4 + 72*x^2 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 15*x^18 + 86*x^16 - 222*x^14 + 158*x^12 + 468*x^10 - 1211*x^8 + 1137*x^6 - 474*x^4 + 72*x^2 - 1)
\( x^{20} - 15 x^{18} + 86 x^{16} - 222 x^{14} + 158 x^{12} + 468 x^{10} - 1211 x^{8} + 1137 x^{6} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[18, 1]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-2616374988854477216011959402496\)
\(\medspace = -\,2^{20}\cdot 11^{16}\cdot 7369^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(33.18\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(11\), \(7369\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{23}a^{16}+\frac{11}{23}a^{14}+\frac{6}{23}a^{12}+\frac{2}{23}a^{10}-\frac{8}{23}a^{8}+\frac{11}{23}a^{6}+\frac{2}{23}a^{4}-\frac{8}{23}a^{2}-\frac{11}{23}$, $\frac{1}{23}a^{17}+\frac{11}{23}a^{15}+\frac{6}{23}a^{13}+\frac{2}{23}a^{11}-\frac{8}{23}a^{9}+\frac{11}{23}a^{7}+\frac{2}{23}a^{5}-\frac{8}{23}a^{3}-\frac{11}{23}a$, $\frac{1}{23}a^{18}+\frac{5}{23}a^{12}-\frac{7}{23}a^{10}+\frac{7}{23}a^{8}-\frac{4}{23}a^{6}-\frac{7}{23}a^{4}+\frac{8}{23}a^{2}+\frac{6}{23}$, $\frac{1}{23}a^{19}+\frac{5}{23}a^{13}-\frac{7}{23}a^{11}+\frac{7}{23}a^{9}-\frac{4}{23}a^{7}-\frac{7}{23}a^{5}+\frac{8}{23}a^{3}+\frac{6}{23}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $18$ | 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: |
$\frac{27}{23}a^{16}-\frac{324}{23}a^{14}+\frac{1312}{23}a^{12}-\frac{1602}{23}a^{10}-\frac{2378}{23}a^{8}+\frac{7680}{23}a^{6}-\frac{6133}{23}a^{4}+\frac{1578}{23}a^{2}-\frac{67}{23}$, $\frac{2}{23}a^{16}-\frac{24}{23}a^{14}+\frac{104}{23}a^{12}-\frac{180}{23}a^{10}-\frac{39}{23}a^{8}+\frac{666}{23}a^{6}-\frac{962}{23}a^{4}+\frac{375}{23}a^{2}+\frac{1}{23}$, $\frac{8}{23}a^{16}-\frac{96}{23}a^{14}+\frac{393}{23}a^{12}-\frac{513}{23}a^{10}-\frac{616}{23}a^{8}+\frac{2319}{23}a^{6}-\frac{2123}{23}a^{4}+\frac{672}{23}a^{2}-\frac{42}{23}$, $\frac{6}{23}a^{16}-\frac{72}{23}a^{14}+\frac{289}{23}a^{12}-\frac{333}{23}a^{10}-\frac{577}{23}a^{8}+\frac{1653}{23}a^{6}-\frac{1161}{23}a^{4}+\frac{297}{23}a^{2}-\frac{43}{23}$, $\frac{28}{23}a^{19}-\frac{384}{23}a^{17}+\frac{1940}{23}a^{15}-\frac{4050}{23}a^{13}+\frac{692}{23}a^{11}+\frac{11617}{23}a^{9}-\frac{20367}{23}a^{7}+\frac{14745}{23}a^{5}-\frac{4869}{23}a^{3}+\frac{620}{23}a$, $\frac{27}{23}a^{17}-\frac{324}{23}a^{15}+\frac{1312}{23}a^{13}-\frac{1602}{23}a^{11}-\frac{2378}{23}a^{9}+\frac{7680}{23}a^{7}-\frac{6133}{23}a^{5}+\frac{1578}{23}a^{3}-\frac{67}{23}a$, $\frac{30}{23}a^{19}-\frac{393}{23}a^{17}+\frac{1841}{23}a^{15}-141a^{13}-\frac{1180}{23}a^{11}+\frac{11841}{23}a^{9}-\frac{15000}{23}a^{7}+\frac{6686}{23}a^{5}-\frac{526}{23}a^{3}-\frac{143}{23}a$, $\frac{2}{23}a^{18}-\frac{26}{23}a^{16}+\frac{128}{23}a^{14}-\frac{284}{23}a^{12}+\frac{141}{23}a^{10}+\frac{705}{23}a^{8}-\frac{1628}{23}a^{6}+\frac{1337}{23}a^{4}-\frac{397}{23}a^{2}+\frac{45}{23}$, $\frac{5}{23}a^{19}-\frac{73}{23}a^{17}+\frac{393}{23}a^{15}-\frac{850}{23}a^{13}-\frac{20}{23}a^{11}+\frac{3057}{23}a^{9}-\frac{4204}{23}a^{7}+\frac{1015}{23}a^{5}+\frac{831}{23}a^{3}-\frac{202}{23}a$, $\frac{2}{23}a^{19}-\frac{2}{23}a^{18}-\frac{28}{23}a^{17}+\frac{28}{23}a^{16}+\frac{152}{23}a^{15}-\frac{152}{23}a^{14}-\frac{388}{23}a^{13}+\frac{388}{23}a^{12}+\frac{321}{23}a^{11}-\frac{321}{23}a^{10}+\frac{744}{23}a^{9}-\frac{744}{23}a^{8}-\frac{2294}{23}a^{7}+\frac{2294}{23}a^{6}+\frac{2299}{23}a^{5}-\frac{2299}{23}a^{4}-\frac{772}{23}a^{3}+\frac{772}{23}a^{2}+\frac{44}{23}a-\frac{44}{23}$, $\frac{1}{23}a^{19}-\frac{15}{23}a^{17}+\frac{88}{23}a^{15}-\frac{246}{23}a^{13}+\frac{262}{23}a^{11}+\frac{288}{23}a^{9}-\frac{1250}{23}a^{7}+\frac{1803}{23}a^{5}-\frac{1436}{23}a^{3}+\frac{447}{23}a-1$, $\frac{36}{23}a^{19}+\frac{2}{23}a^{18}-\frac{474}{23}a^{17}-\frac{26}{23}a^{16}+\frac{2238}{23}a^{15}+\frac{128}{23}a^{14}-\frac{4021}{23}a^{13}-\frac{284}{23}a^{12}-\frac{1154}{23}a^{11}+\frac{141}{23}a^{10}+\frac{14118}{23}a^{9}+\frac{705}{23}a^{8}-\frac{18744}{23}a^{7}-\frac{1628}{23}a^{6}+\frac{9564}{23}a^{5}+\frac{1337}{23}a^{4}-\frac{1716}{23}a^{3}-\frac{397}{23}a^{2}+\frac{117}{23}a+\frac{22}{23}$, $\frac{3}{23}a^{19}-\frac{47}{23}a^{17}-\frac{30}{23}a^{16}+\frac{265}{23}a^{15}+\frac{360}{23}a^{14}-\frac{566}{23}a^{13}-\frac{1445}{23}a^{12}-7a^{11}+\frac{1665}{23}a^{10}+\frac{2352}{23}a^{9}+\frac{2908}{23}a^{8}-112a^{7}-\frac{8403}{23}a^{6}-14a^{5}+\frac{5874}{23}a^{4}+\frac{1251}{23}a^{3}-\frac{1071}{23}a^{2}-\frac{293}{23}a-\frac{15}{23}$, $\frac{36}{23}a^{18}-\frac{27}{23}a^{17}-\frac{468}{23}a^{16}+\frac{324}{23}a^{15}+\frac{2166}{23}a^{14}-\frac{1312}{23}a^{13}-\frac{3732}{23}a^{12}+\frac{1602}{23}a^{11}-\frac{1487}{23}a^{10}+\frac{2378}{23}a^{9}+\frac{13541}{23}a^{8}-\frac{7680}{23}a^{7}-\frac{17091}{23}a^{6}+\frac{6133}{23}a^{5}+\frac{8403}{23}a^{4}-\frac{1578}{23}a^{3}-\frac{1419}{23}a^{2}+\frac{67}{23}a+\frac{51}{23}$, $\frac{1}{23}a^{19}-\frac{15}{23}a^{17}-\frac{9}{23}a^{16}+\frac{88}{23}a^{15}+\frac{108}{23}a^{14}-\frac{246}{23}a^{13}-\frac{422}{23}a^{12}+\frac{262}{23}a^{11}+\frac{396}{23}a^{10}+\frac{288}{23}a^{9}+\frac{1107}{23}a^{8}-\frac{1250}{23}a^{7}-\frac{2376}{23}a^{6}+\frac{1803}{23}a^{5}+\frac{902}{23}a^{4}-\frac{1436}{23}a^{3}+\frac{210}{23}a^{2}+\frac{447}{23}a-\frac{16}{23}$, $\frac{5}{23}a^{19}+\frac{32}{23}a^{18}-\frac{43}{23}a^{17}-\frac{416}{23}a^{16}+\frac{33}{23}a^{15}+\frac{1933}{23}a^{14}+\frac{595}{23}a^{13}-\frac{3394}{23}a^{12}-\frac{1685}{23}a^{11}-\frac{1102}{23}a^{10}+\frac{149}{23}a^{9}+\frac{12016}{23}a^{8}+\frac{4199}{23}a^{7}-\frac{15905}{23}a^{6}-\frac{4859}{23}a^{5}+\frac{8282}{23}a^{4}+\frac{1925}{23}a^{3}-\frac{1453}{23}a^{2}-\frac{256}{23}a+\frac{7}{23}$, $\frac{36}{23}a^{19}-\frac{2}{23}a^{18}-21a^{17}+\frac{35}{23}a^{16}+102a^{15}-\frac{236}{23}a^{14}-\frac{4443}{23}a^{13}+\frac{706}{23}a^{12}-\frac{758}{23}a^{11}-\frac{537}{23}a^{10}+\frac{15225}{23}a^{9}-\frac{1812}{23}a^{8}-\frac{21120}{23}a^{7}+\frac{4004}{23}a^{6}+\frac{10466}{23}a^{5}-\frac{2239}{23}a^{4}-\frac{1506}{23}a^{3}+\frac{187}{23}a^{2}+\frac{124}{23}a-\frac{29}{23}$, $\frac{28}{23}a^{19}-\frac{390}{23}a^{17}+\frac{3}{23}a^{16}+\frac{2012}{23}a^{15}-\frac{36}{23}a^{14}-\frac{4339}{23}a^{13}+\frac{133}{23}a^{12}+\frac{1025}{23}a^{11}-\frac{63}{23}a^{10}+\frac{12194}{23}a^{9}-\frac{530}{23}a^{8}-\frac{22020}{23}a^{7}+\frac{723}{23}a^{6}+\frac{15906}{23}a^{5}+\frac{259}{23}a^{4}-\frac{5166}{23}a^{3}-\frac{507}{23}a^{2}+\frac{686}{23}a+\frac{82}{23}$
| 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: | \( 313694053.017 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{1}\cdot 313694053.017 \cdot 1}{2\cdot\sqrt{2616374988854477216011959402496}}\cr\approx \mathstrut & 0.159715139698 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 15*x^18 + 86*x^16 - 222*x^14 + 158*x^12 + 468*x^10 - 1211*x^8 + 1137*x^6 - 474*x^4 + 72*x^2 - 1)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^20 - 15*x^18 + 86*x^16 - 222*x^14 + 158*x^12 + 468*x^10 - 1211*x^8 + 1137*x^6 - 474*x^4 + 72*x^2 - 1, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^20 - 15*x^18 + 86*x^16 - 222*x^14 + 158*x^12 + 468*x^10 - 1211*x^8 + 1137*x^6 - 474*x^4 + 72*x^2 - 1);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 15*x^18 + 86*x^16 - 222*x^14 + 158*x^12 + 468*x^10 - 1211*x^8 + 1137*x^6 - 474*x^4 + 72*x^2 - 1);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
$C_2^{10}.C_2\wr C_5$ (as 20T846):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A solvable group of order 163840 |
| The 649 conjugacy class representatives for $C_2^{10}.C_2\wr C_5$ |
| Character table for $C_2^{10}.C_2\wr C_5$ |
Intermediate fields
| \(\Q(\zeta_{11})^+\), 10.10.1579610594089.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Sibling fields
| Degree 20 siblings: | data not computed |
| Minimal sibling: | 20.8.355051565864361136655171584.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | $20$ | R | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | $20$ | ${\href{/padicField/37.10.0.1}{10} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{8}$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.10.10.9 | $x^{10} + 10 x^{9} + 22 x^{8} - 8 x^{7} + 72 x^{6} + 1328 x^{5} + 4496 x^{4} + 3680 x^{3} - 3696 x^{2} - 96 x - 32$ | $2$ | $5$ | $10$ | $C_2 \times (C_2^4 : C_5)$ | $[2, 2, 2, 2, 2]^{5}$ |
| 2.10.10.3 | $x^{10} + 10 x^{9} + 74 x^{8} + 320 x^{7} + 1104 x^{6} + 2752 x^{5} + 6176 x^{4} + 12096 x^{3} + 17712 x^{2} + 15968 x + 8416$ | $2$ | $5$ | $10$ | $C_2^4 : C_5$ | $[2, 2, 2, 2]^{5}$ | |
|
\(11\)
| 11.20.16.1 | $x^{20} + 40 x^{18} + 50 x^{17} + 650 x^{16} + 1644 x^{15} + 6440 x^{14} + 18720 x^{13} + 39010 x^{12} + 137400 x^{11} + 283734 x^{10} + 677980 x^{9} + 845540 x^{8} + 1499320 x^{7} + 2045360 x^{6} + 2492700 x^{5} + 5064740 x^{4} + 1884960 x^{3} + 9207500 x^{2} - 2109440 x + 7196441$ | $5$ | $4$ | $16$ | 20T1 | $[\ ]_{5}^{4}$ |
|
\(7369\)
| $\Q_{7369}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{7369}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{7369}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{7369}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{7369}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{7369}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{7369}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{7369}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |