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)

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

Mordell-Weil group structure

trivial

magma: MordellWeilGroup(E);
 

Invariants

Conductor: NN  =  550 550  = 252112 \cdot 5^{2} \cdot 11
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: Δ\Delta  =  563200-563200 = 12115211-1 \cdot 2^{11} \cdot 5^{2} \cdot 11
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: jj  =  3840177158522528 -\frac{38401771585}{22528}  = 1211511119733-1 \cdot 2^{-11} \cdot 5 \cdot 11^{-1} \cdot 1973^{3}
comment: j-invariant
 
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
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)
sage: E.has_cm()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: ST(E)\mathrm{ST}(E) = SU(2)\mathrm{SU}(2)
Faltings height: hFaltingsh_{\mathrm{Faltings}} ≈ 0.0484334737378592367256510553460.048433473737859236725651055346
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: hstableh_{\mathrm{stable}} ≈ 0.21980617833449082570780883353-0.21980617833449082570780883353
magma: StableFaltingsHeight(E);
 
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
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Mordell-Weil rank: rr = 0 0
comment: Rank
 
sage: E.rank()
 
gp: [lower,upper] = ellrank(E)
 
magma: Rank(E);
 
Regulator: Reg(E/Q)\mathrm{Reg}(E/\Q) = 11
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: Ω\Omega ≈ 0.630434998838431966939963744010.63043499883843196693996374401
comment: Real Period
 
sage: E.period_lattice().omega()
 
gp: if(E.disc>0,2,1)*E.omega[1]
 
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
 
Tamagawa product: pcp\prod_{p}c_p = 1 1
comment: Tamagawa numbers
 
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
magma: TamagawaNumbers(E);
 
oscar: tamagawa_numbers(E)
 
Torsion order: #E(Q)tor\#E(\Q)_{\mathrm{tor}} = 11
comment: Torsion order
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
oscar: prod(torsion_structure(E)[1])
 
Special value: L(E,1) L(E,1) ≈ 0.630434998838431966939963744000.63043499883843196693996374400
comment: Special L-value
 
r = E.rank();
 
E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
 
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
Analytic order of Ш: Шan{}_{\mathrm{an}}  =  11    (exact)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 

BSD formula

0.630434999L(E,1)=#Ш(E/Q)ΩEReg(E/Q)pcp#E(Q)tor210.6304351.0000001120.630434999\displaystyle 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

# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)
 
E = EllipticCurve(%s); 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)))
 
/* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */
 
E := EllipticCurve(%s); 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

comment: q-expansion of modular form
 
sage: E.q_eigenform(20)
 
\\ 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
 
magma: ModularForm(E);
 

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

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

comment: Local data
 
sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
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.

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

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)
 
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

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.