LMFDB - Number field 21.13.8866863017363332031653762708238323679232.1

Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

comment:Define the number field

sage:x = polygen(QQ); K.<a> = NumberField(x^21 + 24*x^19 - 16*x^18 - 36*x^17 + 48*x^16 - 3472*x^15 + 6912*x^14 - 11088*x^13 + 18304*x^12 + 107136*x^11 - 407040*x^10 + 691648*x^9 - 928512*x^8 - 63744*x^7 + 4379648*x^6 - 10036224*x^5 + 11501568*x^4 - 7729152*x^3 + 3096576*x^2 - 688128*x + 65536)

gp:K = bnfinit(y^21 + 24*y^19 - 16*y^18 - 36*y^17 + 48*y^16 - 3472*y^15 + 6912*y^14 - 11088*y^13 + 18304*y^12 + 107136*y^11 - 407040*y^10 + 691648*y^9 - 928512*y^8 - 63744*y^7 + 4379648*y^6 - 10036224*y^5 + 11501568*y^4 - 7729152*y^3 + 3096576*y^2 - 688128*y + 65536, 1)

magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 + 24*x^19 - 16*x^18 - 36*x^17 + 48*x^16 - 3472*x^15 + 6912*x^14 - 11088*x^13 + 18304*x^12 + 107136*x^11 - 407040*x^10 + 691648*x^9 - 928512*x^8 - 63744*x^7 + 4379648*x^6 - 10036224*x^5 + 11501568*x^4 - 7729152*x^3 + 3096576*x^2 - 688128*x + 65536);

oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^21 + 24*x^19 - 16*x^18 - 36*x^17 + 48*x^16 - 3472*x^15 + 6912*x^14 - 11088*x^13 + 18304*x^12 + 107136*x^11 - 407040*x^10 + 691648*x^9 - 928512*x^8 - 63744*x^7 + 4379648*x^6 - 10036224*x^5 + 11501568*x^4 - 7729152*x^3 + 3096576*x^2 - 688128*x + 65536)

$ x^{21} + 24 x^{19} - 16 x^{18} - 36 x^{17} + 48 x^{16} - 3472 x^{15} + 6912 x^{14} - 11088 x^{13} + \cdots + 65536 $ x^21 + 24*x^19 - 16*x^18 - 36*x^17 + 48*x^16 - 3472*x^15 + 6912*x^14 - 11088*x^13 + 18304*x^12 + 107136*x^11 - 407040*x^10 + 691648*x^9 - 928512*x^8 - 63744*x^7 + 4379648*x^6 - 10036224*x^5 + 11501568*x^4 - 7729152*x^3 + 3096576*x^2 - 688128*x + 65536

comment:Defining polynomial

sage:K.defining_polynomial()

gp:K.pol

magma:DefiningPolynomial(K);

oscar:defining_polynomial(K)

Invariants

Degree:	$21$	comment:Degree over Q sage:K.degree() gp:poldegree(K.pol) magma:Degree(K); oscar:degree(K)
Signature:	$[13, 4]$	comment:Signature sage:K.signature() gp:K.sign magma:Signature(K); oscar:signature(K)
Discriminant:	$8866863017363332031653762708238323679232$ 8866863017363332031653762708238323679232 $\medspace = 2^{38}\cdot 3^{21}\cdot 11\cdot 809^{6}$	comment:Discriminant sage:K.disc() gp:K.disc magma:OK := Integers(K); Discriminant(OK); oscar:OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$79.85$	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:	$2$, $3$, $11$, $809$ 2, 3, 11, 809	comment:Ramified primes sage:K.disc().support() gp:factor(abs(K.disc))[,1]~ magma:PrimeDivisors(Discriminant(OK)); oscar:prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{33}) $
$\Aut(K/\Q)$:	$C_1$	comment:Autmorphisms sage:K.automorphisms() magma:Automorphisms(K); oscar:automorphisms(K)
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

Integral basis (with respect to field generator $a$)

$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{16}a^{7}-\frac{1}{8}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{16}a^{8}$, $\frac{1}{32}a^{9}-\frac{1}{2}a$, $\frac{1}{32}a^{10}$, $\frac{1}{64}a^{11}-\frac{1}{4}a^{3}$, $\frac{1}{64}a^{12}$, $\frac{1}{128}a^{13}-\frac{1}{8}a^{5}$, $\frac{1}{256}a^{14}-\frac{1}{64}a^{10}-\frac{1}{16}a^{6}-\frac{1}{4}a^{2}$, $\frac{1}{512}a^{15}-\frac{1}{128}a^{11}-\frac{1}{32}a^{7}-\frac{1}{8}a^{3}$, $\frac{1}{2048}a^{16}-\frac{1}{1024}a^{15}-\frac{1}{512}a^{14}-\frac{1}{256}a^{13}-\frac{1}{512}a^{12}-\frac{1}{256}a^{11}-\frac{1}{128}a^{8}+\frac{1}{64}a^{7}+\frac{1}{32}a^{6}+\frac{1}{16}a^{5}-\frac{1}{32}a^{4}-\frac{1}{16}a^{3}$, $\frac{1}{8192}a^{17}-\frac{1}{512}a^{14}+\frac{3}{2048}a^{13}+\frac{3}{512}a^{12}-\frac{3}{512}a^{11}-\frac{1}{64}a^{10}+\frac{7}{512}a^{9}+\frac{1}{64}a^{8}-\frac{1}{32}a^{7}-\frac{1}{16}a^{6}+\frac{11}{128}a^{5}-\frac{3}{32}a^{4}-\frac{3}{32}a^{3}-\frac{1}{8}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{32768}a^{18}-\frac{1}{16384}a^{17}-\frac{1}{2048}a^{15}-\frac{5}{8192}a^{14}+\frac{3}{4096}a^{13}-\frac{1}{2048}a^{12}+\frac{7}{1024}a^{11}+\frac{23}{2048}a^{10}-\frac{3}{1024}a^{9}+\frac{1}{64}a^{8}+\frac{11}{512}a^{6}-\frac{1}{256}a^{5}-\frac{5}{128}a^{4}+\frac{9}{64}a^{3}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{131072}a^{19}-\frac{1}{32768}a^{17}-\frac{1}{8192}a^{16}+\frac{19}{32768}a^{15}-\frac{1}{8192}a^{14}-\frac{15}{4096}a^{13}-\frac{13}{2048}a^{12}+\frac{19}{8192}a^{11}-\frac{11}{1024}a^{10}+\frac{5}{2048}a^{9}+\frac{1}{128}a^{8}+\frac{43}{2048}a^{7}+\frac{5}{512}a^{6}-\frac{19}{256}a^{5}-\frac{7}{64}a^{4}+\frac{1}{128}a^{3}+\frac{1}{16}a^{2}+\frac{5}{16}a-\frac{1}{8}$, $\frac{1}{524288}a^{20}-\frac{1}{262144}a^{19}-\frac{1}{131072}a^{18}-\frac{1}{65536}a^{17}+\frac{27}{131072}a^{16}+\frac{43}{65536}a^{15}-\frac{7}{8192}a^{14}+\frac{1}{4096}a^{13}+\frac{251}{32768}a^{12}-\frac{127}{16384}a^{11}-\frac{15}{8192}a^{10}+\frac{3}{4096}a^{9}-\frac{117}{8192}a^{8}+\frac{31}{4096}a^{7}+\frac{1}{128}a^{6}-\frac{59}{512}a^{5}-\frac{35}{512}a^{4}-\frac{13}{256}a^{3}+\frac{3}{64}a^{2}-\frac{3}{16}a-\frac{7}{16}$ 1, a, 1/2*a^2, 1/2*a^3, 1/4*a^4, 1/4*a^5, 1/8*a^6, 1/16*a^7 - 1/8*a^5 - 1/4*a^3 - 1/2*a, 1/16*a^8, 1/32*a^9 - 1/2*a, 1/32*a^10, 1/64*a^11 - 1/4*a^3, 1/64*a^12, 1/128*a^13 - 1/8*a^5, 1/256*a^14 - 1/64*a^10 - 1/16*a^6 - 1/4*a^2, 1/512*a^15 - 1/128*a^11 - 1/32*a^7 - 1/8*a^3, 1/2048*a^16 - 1/1024*a^15 - 1/512*a^14 - 1/256*a^13 - 1/512*a^12 - 1/256*a^11 - 1/128*a^8 + 1/64*a^7 + 1/32*a^6 + 1/16*a^5 - 1/32*a^4 - 1/16*a^3, 1/8192*a^17 - 1/512*a^14 + 3/2048*a^13 + 3/512*a^12 - 3/512*a^11 - 1/64*a^10 + 7/512*a^9 + 1/64*a^8 - 1/32*a^7 - 1/16*a^6 + 11/128*a^5 - 3/32*a^4 - 3/32*a^3 - 1/8*a^2 - 1/4*a - 1/2, 1/32768*a^18 - 1/16384*a^17 - 1/2048*a^15 - 5/8192*a^14 + 3/4096*a^13 - 1/2048*a^12 + 7/1024*a^11 + 23/2048*a^10 - 3/1024*a^9 + 1/64*a^8 + 11/512*a^6 - 1/256*a^5 - 5/128*a^4 + 9/64*a^3 + 1/4*a - 1/4, 1/131072*a^19 - 1/32768*a^17 - 1/8192*a^16 + 19/32768*a^15 - 1/8192*a^14 - 15/4096*a^13 - 13/2048*a^12 + 19/8192*a^11 - 11/1024*a^10 + 5/2048*a^9 + 1/128*a^8 + 43/2048*a^7 + 5/512*a^6 - 19/256*a^5 - 7/64*a^4 + 1/128*a^3 + 1/16*a^2 + 5/16*a - 1/8, 1/524288*a^20 - 1/262144*a^19 - 1/131072*a^18 - 1/65536*a^17 + 27/131072*a^16 + 43/65536*a^15 - 7/8192*a^14 + 1/4096*a^13 + 251/32768*a^12 - 127/16384*a^11 - 15/8192*a^10 + 3/4096*a^9 - 117/8192*a^8 + 31/4096*a^7 + 1/128*a^6 - 59/512*a^5 - 35/512*a^4 - 13/256*a^3 + 3/64*a^2 - 3/16*a - 7/16

comment:Integral basis

sage:K.integral_basis()

gp:K.zk

magma:IntegralBasis(K);

oscar:basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$

Class group and class number

Ideal class group:		Trivial group, which has order $1$ (assuming GRH)	comment:Class group sage:K.class_group().invariants() gp:K.clgp magma:ClassGroup(K); oscar:class_group(K)
Narrow class group:		$C_{2}$, which has order $2$ (assuming GRH)	comment:Narrow class group sage:K.narrow_class_group().invariants() gp:bnfnarrow(K) magma:NarrowClassGroup(K);

Unit group

comment:Unit group

sage:UK = K.unit_group()

magma:UK, fUK := UnitGroup(K);

oscar:UK, fUK = unit_group(OK)

Rank:	$16$	comment:Unit rank sage:UK.rank() gp:K.fu magma:UnitRank(K); oscar:rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	comment:Generator for roots of unity 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:	$\frac{530909559}{524288}a^{20}+\frac{173781207}{262144}a^{19}+\frac{3242321541}{131072}a^{18}-\frac{531441}{65536}a^{17}-\frac{4779367011}{131072}a^{16}+\frac{1620875367}{65536}a^{15}-\frac{28669175235}{8192}a^{14}+\frac{19284906951}{4096}a^{13}-\frac{266904170739}{32768}a^{12}+\frac{216309714705}{16384}a^{11}+\frac{959554435779}{8192}a^{10}-\frac{1374209180505}{4096}a^{9}+\frac{3938129258589}{8192}a^{8}-\frac{2562011698173}{4096}a^{7}-\frac{60681527803}{128}a^{6}+\frac{2111829085251}{512}a^{5}-\frac{3820870002873}{512}a^{4}+\frac{1730779511183}{256}a^{3}-\frac{217601174655}{64}a^{2}+\frac{7274895849}{8}a-\frac{1621426507}{16}$, $\frac{2257187391}{524288}a^{20}+\frac{729996543}{262144}a^{19}+\frac{13779200061}{131072}a^{18}-\frac{58054401}{65536}a^{17}-\frac{20390099787}{131072}a^{16}+\frac{6948663759}{65536}a^{15}-\frac{121890580227}{8192}a^{14}+\frac{82467506661}{4096}a^{13}-\frac{1137487418395}{32768}a^{12}+\frac{923232922569}{16384}a^{11}+\frac{4077119283147}{8192}a^{10}-\frac{5859278914745}{4096}a^{9}+\frac{16813537508853}{8192}a^{8}-\frac{10935877434357}{4096}a^{7}-\frac{256179342389}{128}a^{6}+\frac{8991203160375}{512}a^{5}-\frac{16306971425697}{512}a^{4}+\frac{7402443519839}{256}a^{3}-\frac{932614650639}{64}a^{2}+\frac{31245365007}{8}a-\frac{6978981579}{16}$, $\frac{987942771}{524288}a^{20}+\frac{320104053}{262144}a^{19}+\frac{6031366321}{131072}a^{18}-\frac{21662343}{65536}a^{17}-\frac{8919745887}{131072}a^{16}+\frac{3037520425}{65536}a^{15}-\frac{26674916643}{4096}a^{14}+\frac{9015747387}{1024}a^{13}-\frac{497681183887}{32768}a^{12}+\frac{403846389819}{16384}a^{11}+\frac{1784669786271}{8192}a^{10}-\frac{2563408193091}{4096}a^{9}+\frac{7354354391217}{8192}a^{8}-\frac{4783583045979}{4096}a^{7}-\frac{448992441885}{512}a^{6}+\frac{3934487665449}{512}a^{5}-\frac{7133174733645}{512}a^{4}+\frac{3237006703197}{256}a^{3}-\frac{407691248895}{64}a^{2}+\frac{27308813931}{16}a-\frac{3048828617}{16}$, $\frac{1221043653}{262144}a^{20}+\frac{98590365}{32768}a^{19}+\frac{7453646919}{65536}a^{18}-\frac{17382465}{16384}a^{17}-\frac{11033854593}{65536}a^{16}+\frac{1881402649}{16384}a^{15}-\frac{131875650909}{8192}a^{14}+\frac{89280737361}{4096}a^{13}-\frac{615517993713}{16384}a^{12}+\frac{31227976683}{512}a^{11}+\frac{2205391476861}{4096}a^{10}-\frac{12385300795}{8}a^{9}+\frac{9099860891031}{4096}a^{8}-\frac{2959299111657}{1024}a^{7}-\frac{1107745057049}{512}a^{6}+\frac{1216151533017}{64}a^{5}-\frac{8825249329911}{256}a^{4}+\frac{1001801602863}{32}a^{3}-\frac{504991870335}{32}a^{2}+\frac{67693819527}{16}a-472643508$, $\frac{5761093815}{524288}a^{20}+\frac{1866281877}{262144}a^{19}+\frac{35171136981}{131072}a^{18}-\frac{128647143}{65536}a^{17}-\frac{52016542995}{131072}a^{16}+\frac{17716025089}{65536}a^{15}-\frac{155552290803}{4096}a^{14}+\frac{52579514481}{1024}a^{13}-\frac{2902331747299}{32768}a^{12}+\frac{2355147605835}{16384}a^{11}+\frac{10407011108427}{8192}a^{10}-\frac{14948971282795}{4096}a^{9}+\frac{42889517808093}{8192}a^{8}-\frac{27897097277331}{4096}a^{7}-\frac{2617914649029}{512}a^{6}+\frac{22944073595013}{512}a^{5}-\frac{41599210091265}{512}a^{4}+\frac{18878415365109}{256}a^{3}-\frac{2377800740619}{64}a^{2}+\frac{159284132937}{16}a-\frac{17784130737}{16}$, $\frac{558665451}{131072}a^{20}+\frac{181598211}{65536}a^{19}+\frac{3411025795}{32768}a^{18}-\frac{8550171}{16384}a^{17}-\frac{5039022327}{32768}a^{16}+\frac{1714054499}{16384}a^{15}-\frac{60336627219}{4096}a^{14}+\frac{40723017075}{2048}a^{13}-\frac{281259523599}{8192}a^{12}+\frac{228132037629}{4096}a^{11}+\frac{1009360582101}{2048}a^{10}-\frac{1448455411599}{1024}a^{9}+\frac{4154199295401}{2048}a^{8}-\frac{2702231363649}{1024}a^{7}-\frac{508736175901}{256}a^{6}+\frac{139002420933}{8}a^{5}-\frac{4029610493379}{128}a^{4}+\frac{1827633757157}{64}a^{3}-\frac{230065672059}{16}a^{2}+\frac{7701435855}{2}a-\frac{1718749045}{4}$, $\frac{1770030643}{65536}a^{20}+\frac{1130388235}{65536}a^{19}+\frac{21601330241}{32768}a^{18}-\frac{91261853}{8192}a^{17}-\frac{16046655347}{16384}a^{16}+\frac{10992665613}{16384}a^{15}-\frac{764683292237}{8192}a^{14}+\frac{520479796987}{4096}a^{13}-\frac{894245439041}{4096}a^{12}+\frac{1453830923933}{4096}a^{11}+\frac{6390284030363}{2048}a^{10}-\frac{4608444418153}{512}a^{9}+\frac{13242628510625}{1024}a^{8}-\frac{17222630052011}{1024}a^{7}-\frac{6380815831721}{512}a^{6}+\frac{28244157389595}{256}a^{5}-\frac{25677450723233}{128}a^{4}+\frac{5840969643027}{32}a^{3}-92186614891a^{2}+\frac{198101648751}{8}a-\frac{5543468093}{2}$, $\frac{7449985813}{524288}a^{20}+\frac{2425723765}{262144}a^{19}+\frac{45489698611}{131072}a^{18}-\frac{88496783}{65536}a^{17}-\frac{67166005401}{131072}a^{16}+\frac{22830313605}{65536}a^{15}-\frac{201151639985}{4096}a^{14}+\frac{135654412209}{2048}a^{13}-\frac{3749394933721}{32768}a^{12}+\frac{3040573916819}{16384}a^{11}+\frac{13461304590213}{8192}a^{10}-\frac{19307948670463}{4096}a^{9}+\frac{55364974699175}{8192}a^{8}-\frac{36015022663111}{4096}a^{7}-\frac{3395437186797}{512}a^{6}+\frac{29652529875883}{512}a^{5}-\frac{53707417684239}{512}a^{4}+\frac{24351680087993}{256}a^{3}-\frac{3064508062101}{64}a^{2}+\frac{102552659547}{8}a-\frac{22879676253}{16}$, $\frac{72296629}{32768}a^{20}+\frac{184522555}{131072}a^{19}+\frac{1764559417}{32768}a^{18}-\frac{30819005}{32768}a^{17}-\frac{163887797}{2048}a^{16}+\frac{1797150785}{32768}a^{15}-\frac{15616708757}{2048}a^{14}+\frac{21267522307}{2048}a^{13}-\frac{36532493117}{2048}a^{12}+\frac{237591415033}{8192}a^{11}+\frac{521996235709}{2048}a^{10}-\frac{1506150436239}{2048}a^{9}+\frac{541061058131}{512}a^{8}-\frac{2814660882439}{2048}a^{7}-\frac{260484888913}{256}a^{6}+\frac{1153736872175}{128}a^{5}-\frac{2098162184279}{128}a^{4}+\frac{1909438946797}{128}a^{3}-\frac{120566433325}{16}a^{2}+\frac{32392267851}{16}a-\frac{1813253005}{8}$, $\frac{103521073}{262144}a^{20}+\frac{17350959}{65536}a^{19}+\frac{632732119}{65536}a^{18}+\frac{1260477}{8192}a^{17}-\frac{925611917}{65536}a^{16}+\frac{38827163}{4096}a^{15}-\frac{11179892641}{8192}a^{14}+\frac{7432582631}{4096}a^{13}-\frac{51784690253}{16384}a^{12}+\frac{20923563471}{4096}a^{11}+\frac{187333079173}{4096}a^{10}-\frac{133203630219}{1024}a^{9}+\frac{761331838843}{4096}a^{8}-\frac{123880189571}{512}a^{7}-\frac{96006582209}{512}a^{6}+\frac{205310044783}{128}a^{5}-\frac{739239013975}{256}a^{4}+\frac{166690759131}{64}a^{3}-\frac{41729120539}{32}a^{2}+\frac{5555085961}{16}a-\frac{154037209}{4}$, $\frac{1513189163}{65536}a^{20}+\frac{1959940109}{131072}a^{19}+\frac{18475604623}{32768}a^{18}-\frac{140346847}{32768}a^{17}-\frac{13664009025}{16384}a^{16}+\frac{18618562095}{32768}a^{15}-\frac{326854869285}{4096}a^{14}+\frac{55252538127}{512}a^{13}-\frac{762384110195}{4096}a^{12}+\frac{2474715079071}{8192}a^{11}+\frac{5466820980291}{2048}a^{10}-\frac{15707342667277}{2048}a^{9}+\frac{11266890373641}{1024}a^{8}-\frac{29313571004665}{2048}a^{7}-\frac{1374879888351}{128}a^{6}+\frac{6026701689897}{64}a^{5}-\frac{21855552196189}{128}a^{4}+\frac{19838367749151}{128}a^{3}-\frac{1249452930457}{16}a^{2}+\frac{334820711449}{16}a-\frac{18693023119}{8}$, $\frac{1084305019}{131072}a^{20}+\frac{707036957}{131072}a^{19}+\frac{6621112209}{32768}a^{18}-\frac{19794777}{32768}a^{17}-\frac{9771052291}{32768}a^{16}+\frac{6640726363}{32768}a^{15}-\frac{234211781523}{8192}a^{14}+\frac{157849297519}{4096}a^{13}-\frac{545587886051}{8192}a^{12}+\frac{884699456127}{8192}a^{11}+\frac{1959337332765}{2048}a^{10}-\frac{5618552832079}{2048}a^{9}+\frac{8054604520297}{2048}a^{8}-\frac{10479337133413}{2048}a^{7}-\frac{1978181889405}{512}a^{6}+\frac{8630119129141}{256}a^{5}-\frac{7813645404387}{128}a^{4}+\frac{7084239182061}{128}a^{3}-\frac{111418816977}{4}a^{2}+\frac{119295589715}{16}a-\frac{6652771909}{8}$, $\frac{3940743519}{524288}a^{20}+\frac{1290899771}{262144}a^{19}+\frac{24067051893}{131072}a^{18}+\frac{2166755}{65536}a^{17}-\frac{35470887563}{131072}a^{16}+\frac{12022854147}{65536}a^{15}-\frac{212800264959}{8192}a^{14}+\frac{143091452943}{4096}a^{13}-\frac{1980717211771}{32768}a^{12}+\frac{1605177312381}{16384}a^{11}+\frac{7122738315139}{8192}a^{10}-\frac{10198393380109}{4096}a^{9}+\frac{29222434533141}{8192}a^{8}-\frac{19011286097121}{4096}a^{7}-\frac{450644229107}{128}a^{6}+\frac{15674054983531}{512}a^{5}-\frac{28353493263209}{512}a^{4}+\frac{12841261984699}{256}a^{3}-\frac{1614131694607}{64}a^{2}+\frac{13487841265}{2}a-\frac{12021365975}{16}$, $\frac{294871193}{262144}a^{20}+\frac{103466537}{131072}a^{19}+\frac{1805570665}{65536}a^{18}+\frac{43834131}{32768}a^{17}-\frac{2591346957}{65536}a^{16}+\frac{860266025}{32768}a^{15}-\frac{31842719061}{8192}a^{14}+\frac{20672905359}{4096}a^{13}-\frac{146348011573}{16384}a^{12}+\frac{117326438935}{8192}a^{11}+\frac{534767110239}{4096}a^{10}-\frac{750036770017}{2048}a^{9}+\frac{2134250936499}{4096}a^{8}-\frac{1390419154435}{2048}a^{7}-\frac{280557499251}{512}a^{6}+\frac{581328605399}{128}a^{5}-\frac{2074219563013}{256}a^{4}+\frac{928449879143}{128}a^{3}-\frac{115415631153}{32}a^{2}+\frac{1908150857}{2}a-\frac{841420211}{8}$, $\frac{144749773}{524288}a^{20}+\frac{47288543}{262144}a^{19}+\frac{883999199}{131072}a^{18}-\frac{668773}{65536}a^{17}-\frac{1302305345}{131072}a^{16}+\frac{443520595}{65536}a^{15}-\frac{977065653}{1024}a^{14}+\frac{1315726821}{1024}a^{13}-\frac{72846696209}{32768}a^{12}+\frac{59012467041}{16384}a^{11}+\frac{261568987873}{8192}a^{10}-\frac{374838343297}{4096}a^{9}+\frac{1074878421455}{8192}a^{8}-\frac{699255322057}{4096}a^{7}-\frac{66060317861}{512}a^{6}+\frac{575850695047}{512}a^{5}-\frac{1042594558099}{512}a^{4}+\frac{472664509359}{256}a^{3}-\frac{59485777409}{64}a^{2}+\frac{3982341971}{16}a-\frac{444427187}{16}$, $\frac{13883726485}{524288}a^{20}+\frac{4453874325}{262144}a^{19}+\frac{84731175147}{131072}a^{18}-\frac{585832679}{65536}a^{17}-\frac{125704738521}{131072}a^{16}+\frac{42976647653}{65536}a^{15}-\frac{93718186721}{1024}a^{14}+\frac{127301059555}{1024}a^{13}-\frac{7007812169081}{32768}a^{12}+\frac{5693405644499}{16384}a^{11}+\frac{25067783696781}{8192}a^{10}-\frac{36108544152455}{4096}a^{9}+\frac{103707371553255}{8192}a^{8}-\frac{67443593795111}{4096}a^{7}-\frac{6273180034483}{512}a^{6}+\frac{55355811829519}{512}a^{5}-\frac{100558373789495}{512}a^{4}+\frac{45712218663745}{256}a^{3}-\frac{5767153764757}{64}a^{2}+\frac{193487870399}{8}a-\frac{43279428197}{16}$ 530909559/524288a^20 + 173781207/262144a^19 + 3242321541/131072a^18 - 531441/65536a^17 - 4779367011/131072a^16 + 1620875367/65536a^15 - 28669175235/8192a^14 + 19284906951/4096a^13 - 266904170739/32768a^12 + 216309714705/16384a^11 + 959554435779/8192a^10 - 1374209180505/4096a^9 + 3938129258589/8192a^8 - 2562011698173/4096a^7 - 60681527803/128a^6 + 2111829085251/512a^5 - 3820870002873/512a^4 + 1730779511183/256a^3 - 217601174655/64a^2 + 7274895849/8a - 1621426507/16, 2257187391/524288a^20 + 729996543/262144a^19 + 13779200061/131072a^18 - 58054401/65536a^17 - 20390099787/131072a^16 + 6948663759/65536a^15 - 121890580227/8192a^14 + 82467506661/4096a^13 - 1137487418395/32768a^12 + 923232922569/16384a^11 + 4077119283147/8192a^10 - 5859278914745/4096a^9 + 16813537508853/8192a^8 - 10935877434357/4096a^7 - 256179342389/128a^6 + 8991203160375/512a^5 - 16306971425697/512a^4 + 7402443519839/256a^3 - 932614650639/64a^2 + 31245365007/8a - 6978981579/16, 987942771/524288a^20 + 320104053/262144a^19 + 6031366321/131072a^18 - 21662343/65536a^17 - 8919745887/131072a^16 + 3037520425/65536a^15 - 26674916643/4096a^14 + 9015747387/1024a^13 - 497681183887/32768a^12 + 403846389819/16384a^11 + 1784669786271/8192a^10 - 2563408193091/4096a^9 + 7354354391217/8192a^8 - 4783583045979/4096a^7 - 448992441885/512a^6 + 3934487665449/512a^5 - 7133174733645/512a^4 + 3237006703197/256a^3 - 407691248895/64a^2 + 27308813931/16a - 3048828617/16, 1221043653/262144a^20 + 98590365/32768a^19 + 7453646919/65536a^18 - 17382465/16384a^17 - 11033854593/65536a^16 + 1881402649/16384a^15 - 131875650909/8192a^14 + 89280737361/4096a^13 - 615517993713/16384a^12 + 31227976683/512a^11 + 2205391476861/4096a^10 - 12385300795/8a^9 + 9099860891031/4096a^8 - 2959299111657/1024a^7 - 1107745057049/512a^6 + 1216151533017/64a^5 - 8825249329911/256a^4 + 1001801602863/32a^3 - 504991870335/32a^2 + 67693819527/16a - 472643508, 5761093815/524288a^20 + 1866281877/262144a^19 + 35171136981/131072a^18 - 128647143/65536a^17 - 52016542995/131072a^16 + 17716025089/65536a^15 - 155552290803/4096a^14 + 52579514481/1024a^13 - 2902331747299/32768a^12 + 2355147605835/16384a^11 + 10407011108427/8192a^10 - 14948971282795/4096a^9 + 42889517808093/8192a^8 - 27897097277331/4096a^7 - 2617914649029/512a^6 + 22944073595013/512a^5 - 41599210091265/512a^4 + 18878415365109/256a^3 - 2377800740619/64a^2 + 159284132937/16a - 17784130737/16, 558665451/131072a^20 + 181598211/65536a^19 + 3411025795/32768a^18 - 8550171/16384a^17 - 5039022327/32768a^16 + 1714054499/16384a^15 - 60336627219/4096a^14 + 40723017075/2048a^13 - 281259523599/8192a^12 + 228132037629/4096a^11 + 1009360582101/2048a^10 - 1448455411599/1024a^9 + 4154199295401/2048a^8 - 2702231363649/1024a^7 - 508736175901/256a^6 + 139002420933/8a^5 - 4029610493379/128a^4 + 1827633757157/64a^3 - 230065672059/16a^2 + 7701435855/2a - 1718749045/4, 1770030643/65536a^20 + 1130388235/65536a^19 + 21601330241/32768a^18 - 91261853/8192a^17 - 16046655347/16384a^16 + 10992665613/16384a^15 - 764683292237/8192a^14 + 520479796987/4096a^13 - 894245439041/4096a^12 + 1453830923933/4096a^11 + 6390284030363/2048a^10 - 4608444418153/512a^9 + 13242628510625/1024a^8 - 17222630052011/1024a^7 - 6380815831721/512a^6 + 28244157389595/256a^5 - 25677450723233/128a^4 + 5840969643027/32a^3 - 92186614891a^2 + 198101648751/8a - 5543468093/2, 7449985813/524288a^20 + 2425723765/262144a^19 + 45489698611/131072a^18 - 88496783/65536a^17 - 67166005401/131072a^16 + 22830313605/65536a^15 - 201151639985/4096a^14 + 135654412209/2048a^13 - 3749394933721/32768a^12 + 3040573916819/16384a^11 + 13461304590213/8192a^10 - 19307948670463/4096a^9 + 55364974699175/8192a^8 - 36015022663111/4096a^7 - 3395437186797/512a^6 + 29652529875883/512a^5 - 53707417684239/512a^4 + 24351680087993/256a^3 - 3064508062101/64a^2 + 102552659547/8a - 22879676253/16, 72296629/32768a^20 + 184522555/131072a^19 + 1764559417/32768a^18 - 30819005/32768a^17 - 163887797/2048a^16 + 1797150785/32768a^15 - 15616708757/2048a^14 + 21267522307/2048a^13 - 36532493117/2048a^12 + 237591415033/8192a^11 + 521996235709/2048a^10 - 1506150436239/2048a^9 + 541061058131/512a^8 - 2814660882439/2048a^7 - 260484888913/256a^6 + 1153736872175/128a^5 - 2098162184279/128a^4 + 1909438946797/128a^3 - 120566433325/16a^2 + 32392267851/16a - 1813253005/8, 103521073/262144a^20 + 17350959/65536a^19 + 632732119/65536a^18 + 1260477/8192a^17 - 925611917/65536a^16 + 38827163/4096a^15 - 11179892641/8192a^14 + 7432582631/4096a^13 - 51784690253/16384a^12 + 20923563471/4096a^11 + 187333079173/4096a^10 - 133203630219/1024a^9 + 761331838843/4096a^8 - 123880189571/512a^7 - 96006582209/512a^6 + 205310044783/128a^5 - 739239013975/256a^4 + 166690759131/64a^3 - 41729120539/32a^2 + 5555085961/16a - 154037209/4, 1513189163/65536a^20 + 1959940109/131072a^19 + 18475604623/32768a^18 - 140346847/32768a^17 - 13664009025/16384a^16 + 18618562095/32768a^15 - 326854869285/4096a^14 + 55252538127/512a^13 - 762384110195/4096a^12 + 2474715079071/8192a^11 + 5466820980291/2048a^10 - 15707342667277/2048a^9 + 11266890373641/1024a^8 - 29313571004665/2048a^7 - 1374879888351/128a^6 + 6026701689897/64a^5 - 21855552196189/128a^4 + 19838367749151/128a^3 - 1249452930457/16a^2 + 334820711449/16a - 18693023119/8, 1084305019/131072a^20 + 707036957/131072a^19 + 6621112209/32768a^18 - 19794777/32768a^17 - 9771052291/32768a^16 + 6640726363/32768a^15 - 234211781523/8192a^14 + 157849297519/4096a^13 - 545587886051/8192a^12 + 884699456127/8192a^11 + 1959337332765/2048a^10 - 5618552832079/2048a^9 + 8054604520297/2048a^8 - 10479337133413/2048a^7 - 1978181889405/512a^6 + 8630119129141/256a^5 - 7813645404387/128a^4 + 7084239182061/128a^3 - 111418816977/4a^2 + 119295589715/16a - 6652771909/8, 3940743519/524288a^20 + 1290899771/262144a^19 + 24067051893/131072a^18 + 2166755/65536a^17 - 35470887563/131072a^16 + 12022854147/65536a^15 - 212800264959/8192a^14 + 143091452943/4096a^13 - 1980717211771/32768a^12 + 1605177312381/16384a^11 + 7122738315139/8192a^10 - 10198393380109/4096a^9 + 29222434533141/8192a^8 - 19011286097121/4096a^7 - 450644229107/128a^6 + 15674054983531/512a^5 - 28353493263209/512a^4 + 12841261984699/256a^3 - 1614131694607/64a^2 + 13487841265/2a - 12021365975/16, 294871193/262144a^20 + 103466537/131072a^19 + 1805570665/65536a^18 + 43834131/32768a^17 - 2591346957/65536a^16 + 860266025/32768a^15 - 31842719061/8192a^14 + 20672905359/4096a^13 - 146348011573/16384a^12 + 117326438935/8192a^11 + 534767110239/4096a^10 - 750036770017/2048a^9 + 2134250936499/4096a^8 - 1390419154435/2048a^7 - 280557499251/512a^6 + 581328605399/128a^5 - 2074219563013/256a^4 + 928449879143/128a^3 - 115415631153/32a^2 + 1908150857/2a - 841420211/8, 144749773/524288a^20 + 47288543/262144a^19 + 883999199/131072a^18 - 668773/65536a^17 - 1302305345/131072a^16 + 443520595/65536a^15 - 977065653/1024a^14 + 1315726821/1024a^13 - 72846696209/32768a^12 + 59012467041/16384a^11 + 261568987873/8192a^10 - 374838343297/4096a^9 + 1074878421455/8192a^8 - 699255322057/4096a^7 - 66060317861/512a^6 + 575850695047/512a^5 - 1042594558099/512a^4 + 472664509359/256a^3 - 59485777409/64a^2 + 3982341971/16a - 444427187/16, 13883726485/524288a^20 + 4453874325/262144a^19 + 84731175147/131072a^18 - 585832679/65536a^17 - 125704738521/131072a^16 + 42976647653/65536a^15 - 93718186721/1024a^14 + 127301059555/1024a^13 - 7007812169081/32768a^12 + 5693405644499/16384a^11 + 25067783696781/8192a^10 - 36108544152455/4096a^9 + 103707371553255/8192a^8 - 67443593795111/4096a^7 - 6273180034483/512a^6 + 55355811829519/512a^5 - 100558373789495/512a^4 + 45712218663745/256a^3 - 5767153764757/64a^2 + 193487870399/8a - 43279428197/16 (assuming GRH)	comment:Fundamental units 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:	$ 26033728843700 $ (assuming GRH)	comment:Regulator sage:K.regulator() gp:K.reg magma:Regulator(K); oscar:regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{13}\cdot(2\pi)^{4}\cdot 26033728843700 \cdot 1}{2\cdot\sqrt{8866863017363332031653762708238323679232}}\cr\approx \mathstrut & 1.76494346653112 \end{aligned}\] (assuming GRH)

comment:Analytic class number formula

sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^21 + 24*x^19 - 16*x^18 - 36*x^17 + 48*x^16 - 3472*x^15 + 6912*x^14 - 11088*x^13 + 18304*x^12 + 107136*x^11 - 407040*x^10 + 691648*x^9 - 928512*x^8 - 63744*x^7 + 4379648*x^6 - 10036224*x^5 + 11501568*x^4 - 7729152*x^3 + 3096576*x^2 - 688128*x + 65536) 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))))

gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^21 + 24*x^19 - 16*x^18 - 36*x^17 + 48*x^16 - 3472*x^15 + 6912*x^14 - 11088*x^13 + 18304*x^12 + 107136*x^11 - 407040*x^10 + 691648*x^9 - 928512*x^8 - 63744*x^7 + 4379648*x^6 - 10036224*x^5 + 11501568*x^4 - 7729152*x^3 + 3096576*x^2 - 688128*x + 65536, 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)))]

magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 + 24*x^19 - 16*x^18 - 36*x^17 + 48*x^16 - 3472*x^15 + 6912*x^14 - 11088*x^13 + 18304*x^12 + 107136*x^11 - 407040*x^10 + 691648*x^9 - 928512*x^8 - 63744*x^7 + 4379648*x^6 - 10036224*x^5 + 11501568*x^4 - 7729152*x^3 + 3096576*x^2 - 688128*x + 65536); 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)));

oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 + 24*x^19 - 16*x^18 - 36*x^17 + 48*x^16 - 3472*x^15 + 6912*x^14 - 11088*x^13 + 18304*x^12 + 107136*x^11 - 407040*x^10 + 691648*x^9 - 928512*x^8 - 63744*x^7 + 4379648*x^6 - 10036224*x^5 + 11501568*x^4 - 7729152*x^3 + 3096576*x^2 - 688128*x + 65536); 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

$C_3^7.(C_2^7.\GL(3,2))$ (as 21T147):

comment:Galois group

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 non-solvable group of order 47029248

The 228 conjugacy class representatives for $C_3^7.(C_2^7.\GL(3,2))$

Character table for $C_3^7.(C_2^7.\GL(3,2))$

Intermediate fields

7.7.670188544.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

comment:Intermediate fields

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 42 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	R	${\href{/padicField/5.14.0.1}{14} }{,}\,{\href{/padicField/5.7.0.1}{7} }$	${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.7.0.1}{7} }$	R	${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.7.0.1}{7} }$	$21$	${\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$	${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }$	${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$	${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$	${\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$	$21$	${\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$	${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$	${\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$	${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

comment:Frobenius cycle types

sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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	2.1.3.2a1.1	$x^{3} + 2$	$3$	$1$	$2$	$S_3$	$$[\ ]_{3}^{2}$$
	2.1.6.10a1.6	$x^{6} + 2 x^{5} + 4 x^{4} + 4 x + 2$	$6$	$1$	$10$	$S_4$	$$[\frac{8}{3}, \frac{8}{3}]_{3}^{2}$$
	2.1.12.26a1.29	$x^{12} + 4 x^{6} + 4 x^{5} + 4 x^{3} + 4 x^{2} + 2$	$12$	$1$	$26$	12T100	$$[\frac{4}{3}, \frac{4}{3}, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$$
$3$ 3	3.7.3.21a258.1	$x^{21} + 6 x^{16} + 3 x^{14} + 3 x^{13} + 6 x^{12} + 18 x^{11} + 18 x^{9} + 9 x^{8} + 18 x^{7} + 23 x^{6} + 6 x^{5} + 30 x^{4} + 6 x^{3} + 18 x^{2} + 3 x + 7$	$3$	$7$	$21$	not computed	not computed
$11$ 11	$\Q_{11}$	$x + 9$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	11.1.2.1a1.2	$x^{2} + 22$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	11.3.1.0a1.1	$x^{3} + 2 x + 9$	$1$	$3$	$0$	$C_3$	$$[\ ]^{3}$$
	11.3.1.0a1.1	$x^{3} + 2 x + 9$	$1$	$3$	$0$	$C_3$	$$[\ ]^{3}$$
	11.6.1.0a1.1	$x^{6} + 3 x^{4} + 4 x^{3} + 6 x^{2} + 7 x + 2$	$1$	$6$	$0$	$C_6$	$$[\ ]^{6}$$
	11.6.1.0a1.1	$x^{6} + 3 x^{4} + 4 x^{3} + 6 x^{2} + 7 x + 2$	$1$	$6$	$0$	$C_6$	$$[\ ]^{6}$$
$809$ 809	$\Q_{809}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	$\Q_{809}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	$\Q_{809}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	$\Q_{809}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	$\Q_{809}$	$x$	$1$	$1$	$0$	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 $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $4$	$2$	$2$	$2$

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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
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