Properties

Label 550c1
Conductor 550550
Discriminant 563200-563200
j-invariant 3840177158522528 -\frac{38401771585}{22528}
CM no
Rank 00
Torsion structure trivial

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

y2+xy+y=x3206x1152y^2+xy+y=x^3-206x-1152 Copy content Toggle raw display (homogenize, simplify)
y2z+xyz+yz2=x3206xz21152z3y^2z+xyz+yz^2=x^3-206xz^2-1152z^3 Copy content Toggle raw display (dehomogenize, simplify)
y2=x3266355x52937010y^2=x^3-266355x-52937010 Copy content Toggle raw display (homogenize, minimize)

Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([1, 0, 1, -206, -1152])
 
Copy content gp:E = ellinit([1, 0, 1, -206, -1152])
 
Copy content magma:E := EllipticCurve([1, 0, 1, -206, -1152]);
 
Copy content oscar:E = elliptic_curve([1, 0, 1, -206, -1152])
 
Copy content comment:Simplified equation
 
Copy content sage:E.short_weierstrass_model()
 
Copy content magma:WeierstrassModel(E);
 
Copy content oscar:short_weierstrass_model(E)
 

Mordell-Weil group structure

trivial

Copy content comment:Mordell-Weil group
 
Copy content magma:MordellWeilGroup(E);
 

Invariants

Conductor: NN  =  550 550  = 252112 \cdot 5^{2} \cdot 11
Copy content comment:Conductor
 
Copy content sage:E.conductor().factor()
 
Copy content gp:ellglobalred(E)[1]
 
Copy content magma:Conductor(E);
 
Copy content oscar:conductor(E)
 
Discriminant: Δ\Delta  =  563200-563200 = 12115211-1 \cdot 2^{11} \cdot 5^{2} \cdot 11
Copy content comment:Discriminant
 
Copy content sage:E.discriminant().factor()
 
Copy content gp:E.disc
 
Copy content magma:Discriminant(E);
 
Copy content oscar:discriminant(E)
 
j-invariant: jj  =  3840177158522528 -\frac{38401771585}{22528}  = 1211511119733-1 \cdot 2^{-11} \cdot 5 \cdot 11^{-1} \cdot 1973^{3}
Copy content comment:j-invariant
 
Copy content sage:E.j_invariant().factor()
 
Copy content gp:E.j
 
Copy content magma:jInvariant(E);
 
Copy content oscar:j_invariant(E)
 
Endomorphism ring: End(E)\mathrm{End}(E) = Z\Z
Geometric endomorphism ring: End(EQ)\mathrm{End}(E_{\overline{\Q}})  =  Z\Z    (no potential complex multiplication)
Copy content comment:Potential complex multiplication
 
Copy content sage:E.has_cm()
 
Copy content magma:HasComplexMultiplication(E);
 
Sato-Tate group: ST(E)\mathrm{ST}(E) = SU(2)\mathrm{SU}(2)
Faltings height: hFaltingsh_{\mathrm{Faltings}} ≈ 0.0484334737378592367256510553460.048433473737859236725651055346
Copy content comment:Faltings height
 
Copy content gp:ellheight(E)
 
Copy content magma:FaltingsHeight(E);
 
Copy content oscar:faltings_height(E)
 
Stable Faltings height: hstableh_{\mathrm{stable}} ≈ 0.21980617833449082570780883353-0.21980617833449082570780883353
Copy content comment:Stable Faltings height
 
Copy content magma:StableFaltingsHeight(E);
 
Copy content oscar:stable_faltings_height(E)
 
abcabc quality: QQ ≈ 0.95306648113118180.9530664811311818
Szpiro ratio: σm\sigma_{m} ≈ 4.3726807921349264.372680792134926

BSD invariants

Analytic rank: ranr_{\mathrm{an}} = 0 0
Copy content comment:Analytic rank
 
Copy content sage:E.analytic_rank()
 
Copy content gp:ellanalyticrank(E)
 
Copy content magma:AnalyticRank(E);
 
Mordell-Weil rank: rr = 0 0
Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 
Regulator: Reg(E/Q)\mathrm{Reg}(E/\Q) = 11
Copy content comment:Regulator
 
Copy content sage:E.regulator()
 
Copy content gp:G = E.gen \\ if available matdet(ellheightmatrix(E,G))
 
Copy content magma:Regulator(E);
 
Real period: Ω\Omega ≈ 0.630434998838431966939963744010.63043499883843196693996374401
Copy content comment:Real Period
 
Copy content sage:E.period_lattice().omega()
 
Copy content gp:if(E.disc>0,2,1)*E.omega[1]
 
Copy content magma:(Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
 
Tamagawa product: pcp\prod_{p}c_p = 1 1
Copy content comment:Tamagawa numbers
 
Copy content sage:E.tamagawa_numbers()
 
Copy content gp:gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
Copy content magma:TamagawaNumbers(E);
 
Copy content oscar:tamagawa_numbers(E)
 
Torsion order: #E(Q)tor\#E(\Q)_{\mathrm{tor}} = 11
Copy content comment:Torsion order
 
Copy content sage:E.torsion_order()
 
Copy content gp:elltors(E)[1]
 
Copy content magma:Order(TorsionSubgroup(E));
 
Copy content oscar:prod(torsion_structure(E)[1])
 
Special value: L(E,1) L(E,1) ≈ 0.630434998838431966939963744000.63043499883843196693996374400
Copy content comment:Special L-value
 
Copy content sage:r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
Copy content gp:[r,L1r] = ellanalyticrank(E); L1r/r!
 
Copy content magma:Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
Analytic order of Ш: Шan{}_{\mathrm{an}}  =  11    (exact)
Copy content comment:Order of Sha
 
Copy content sage:E.sha().an_numerical()
 
Copy content magma:MordellWeilShaInformation(E);
 

BSD formula

0.630434999L(E,1)=#Ш(E/Q)ΩEReg(E/Q)pcp#E(Q)tor210.6304351.0000001120.630434999\begin{aligned} 0.630434999 \approx L(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.630435 \cdot 1.000000 \cdot 1}{1^2} \\ & \approx 0.630434999\end{aligned}

Copy content comment:BSD formula
 
Copy content sage:# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha) E = EllipticCurve([1, 0, 1, -206, -1152]); r = E.rank(); ar = E.analytic_rank(); assert r == ar; Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical(); omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order(); assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))
 
Copy content magma:/* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */ E := EllipticCurve([1, 0, 1, -206, -1152]); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar; sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1); reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E); assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);
 

Modular invariants

Modular form   550.2.a.b

qq22q3+q4+2q6q8+q9+q112q123q13+q16+4q17q18q19+O(q20) q - q^{2} - 2 q^{3} + q^{4} + 2 q^{6} - q^{8} + q^{9} + q^{11} - 2 q^{12} - 3 q^{13} + q^{16} + 4 q^{17} - q^{18} - q^{19} + O(q^{20}) Copy content Toggle raw display

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:\\ actual modular form, use for small N [mf,F] = mffromell(E) Ser(mfcoefs(mf,20),q) \\ or just the series Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 

For more coefficients, see the Downloads section to the right.

Modular degree: 132
Copy content comment:Modular degree
 
Copy content sage:E.modular_degree()
 
Copy content gp:ellmoddegree(E)
 
Copy content magma:ModularDegree(E);
 
Γ0(N) \Gamma_0(N) -optimal: yes
Manin constant: 1
Copy content comment:Manin constant
 
Copy content magma:ManinConstant(E);
 

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 3 primes pp of bad reduction:

pp Tamagawa number Kodaira symbol Reduction type Root number ordp(N)\mathrm{ord}_p(N) ordp(Δ)\mathrm{ord}_p(\Delta) ordp(den(j))\mathrm{ord}_p(\mathrm{den}(j))
22 11 I11I_{11} nonsplit multiplicative 1 1 11 11
55 11 IIII additive 1 2 2 0
1111 11 I1I_{1} split multiplicative -1 1 1 1

Copy content comment:Local data
 
Copy content sage:E.local_data()
 
Copy content gp:ellglobalred(E)[5]
 
Copy content magma:[LocalInformation(E,p) : p in BadPrimes(E)];
 
Copy content oscar:[(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]
 

Galois representations

The \ell-adic Galois representation has maximal image for all primes \ell.

Copy content comment:Mod p Galois image
 
Copy content sage:rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]
 
Copy content magma:[GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 

Copy content comment:Adelic image of Galois representation
 
Copy content sage:gens = [[1, 1, 87, 0], [1, 2, 0, 1], [1, 0, 2, 1], [87, 2, 86, 3], [23, 2, 0, 1], [45, 2, 45, 3], [57, 2, 57, 3]] GL(2,Integers(88)).subgroup(gens)
 
Copy content magma:Gens := [[1, 1, 87, 0], [1, 2, 0, 1], [1, 0, 2, 1], [87, 2, 86, 3], [23, 2, 0, 1], [45, 2, 45, 3], [57, 2, 57, 3]]; sub<GL(2,Integers(88))|Gens>;
 

The image H:=ρE(Gal(Q/Q))H:=\rho_E(\Gal(\overline{\Q}/\Q)) of the adelic Galois representation has level 88=2311 88 = 2^{3} \cdot 11 , index 22, genus 00, and generators

(11870),(1201),(1021),(872863),(23201),(452453),(572573)\left(\begin{array}{rr} 1 & 1 \\ 87 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 87 & 2 \\ 86 & 3 \end{array}\right),\left(\begin{array}{rr} 23 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 45 & 2 \\ 45 & 3 \end{array}\right),\left(\begin{array}{rr} 57 & 2 \\ 57 & 3 \end{array}\right).

Input positive integer mm to see the generators of the reduction of HH to GL2(Z/mZ)\mathrm{GL}_2(\Z/m\Z):

The torsion field K:=Q(E[88])K:=\Q(E[88]) is a degree-1013760010137600 Galois extension of Q\Q with Gal(K/Q)\Gal(K/\Q) isomorphic to the projection of HH to GL2(Z/88Z)\GL_2(\Z/88\Z).

The table below list all primes \ell for which the Serre invariants associated to the mod-\ell Galois representation are exceptional.

\ell Reduction type Serre weight Serre conductor
22 nonsplit multiplicative 44 275=5211 275 = 5^{2} \cdot 11
55 additive 1010 22=211 22 = 2 \cdot 11
1111 split multiplicative 1212 25=52 25 = 5^{2}

Isogenies

Copy content comment:Isogenies
 
Copy content gp:ellisomat(E)
 

This curve has no rational isogenies. Its isogeny class 550c consists of this curve only.

Twists

This elliptic curve is its own minimal quadratic twist.

Growth of torsion in number fields

The number fields KK of degree less than 24 such that E(K)torsE(K)_{\rm tors} is strictly larger than E(Q)torsE(\Q)_{\rm tors} (which is trivial) are as follows:

[K:Q][K:\Q] KK E(K)torsE(K)_{\rm tors} Base change curve
33 3.1.2200.1 Z/2Z\Z/2\Z not in database
66 6.0.425920000.1 Z/2ZZ/2Z\Z/2\Z \oplus \Z/2\Z not in database
88 8.2.8004966750000.9 Z/3Z\Z/3\Z not in database
1212 deg 12 Z/4Z\Z/4\Z not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.

Iwasawa invariants

pp 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type nonsplit ord add ss split ord ord ord ord ord ord ord ord ord ord
λ\lambda-invariant(s) 1 6 - 0,0 1 0 0 0 0 0 0 0 0 0 0
μ\mu-invariant(s) 0 0 - 0,0 0 0 0 0 0 0 0 0 0 0 0

An entry - indicates that the invariants are not computed because the reduction is additive.

pp-adic regulators

All pp-adic regulators are identically 11 since the rank is 00.