Properties

Label 1225.b1
Conductor 12251225
Discriminant 6125-6125
j-invariant 162677523113838677 -162677523113838677
CM no
Rank 11
Torsion structure trivial

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

y2+xy+y=x3+x2208083x36621194y^2+xy+y=x^3+x^2-208083x-36621194 Copy content Toggle raw display (homogenize, simplify)
y2z+xyz+yz2=x3+x2z208083xz236621194z3y^2z+xyz+yz^2=x^3+x^2z-208083xz^2-36621194z^3 Copy content Toggle raw display (dehomogenize, simplify)
y2=x3269675595x1704553285050y^2=x^3-269675595x-1704553285050 Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([1, 1, 1, -208083, -36621194])
 
gp: E = ellinit([1, 1, 1, -208083, -36621194])
 
magma: E := EllipticCurve([1, 1, 1, -208083, -36621194]);
 
oscar: E = elliptic_curve([1, 1, 1, -208083, -36621194])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

Z\Z

magma: MordellWeilGroup(E);
 

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(1190,36857)(1190, 36857)6.47720437827251177356138916526.4772043782725117735613891652\infty

Integral points

(1190,36857) \left(1190, 36857\right) , (1190,38048) \left(1190, -38048\right) Copy content Toggle raw display

comment: Integral points
 
sage: E.integral_points()
 
magma: IntegralPoints(E);
 

Invariants

Conductor: NN  =  1225 1225  = 52725^{2} \cdot 7^{2}
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: Δ\Delta  =  6125-6125 = 15372-1 \cdot 5^{3} \cdot 7^{2}
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: jj  =  162677523113838677 -162677523113838677  = 17137320833-1 \cdot 7 \cdot 137^{3} \cdot 2083^{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}} ≈ 1.27048472850736846300063426881.2704847285073684630006342688
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: hstableh_{\mathrm{stable}} ≈ 0.543806892222957818499552311590.54380689222295781849955231159
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
abcabc quality: QQ ≈ 1.07343980001167471.0734398000116747
Szpiro ratio: σm\sigma_{m} ≈ 6.7997112307311456.799711230731145

BSD invariants

Analytic rank: ranr_{\mathrm{an}} = 1 1
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Mordell-Weil rank: rr = 1 1
comment: Rank
 
sage: E.rank()
 
gp: [lower,upper] = ellrank(E)
 
magma: Rank(E);
 
Regulator: Reg(E/Q)\mathrm{Reg}(E/\Q) ≈ 6.47720437827251177356138916526.4772043782725117735613891652
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: Ω\Omega ≈ 0.111766177514525264699392323730.11176617751452526469939232373
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 = 2 2  = 21 2\cdot1
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) ≈ 1.44786474867973160472234629491.4478647486797316047223462949
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    (rounded)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 

BSD formula

1.447864749L(E,1)=#Ш(E/Q)ΩEReg(E/Q)pcp#E(Q)tor210.1117666.4772042121.447864749\displaystyle 1.447864749 \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 0.111766 \cdot 6.477204 \cdot 2}{1^2} \approx 1.447864749

# 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   1225.2.a.b

qq2q3q4+q6+3q82q9+q12+2q13q16+2q17+2q186q19+O(q20) q - q^{2} - q^{3} - q^{4} + q^{6} + 3 q^{8} - 2 q^{9} + q^{12} + 2 q^{13} - q^{16} + 2 q^{17} + 2 q^{18} - 6 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: 1776
comment: Modular degree
 
sage: E.modular_degree()
 
gp: ellmoddegree(E)
 
magma: ModularDegree(E);
 
Γ0(N) \Gamma_0(N) -optimal: no
Manin constant: 1
comment: Manin constant
 
magma: ManinConstant(E);
 

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 2 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))
55 22 IIIIII additive -1 2 3 0
77 11 IIII additive -1 2 2 0

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 except those listed in the table below.

prime \ell mod-\ell image \ell-adic image
3737 37B.8.2 37.114.4.2

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, 3256, 0, 1], [1, 0, 148, 1], [1, 28, 0, 1], [1037, 0, 0, 2073], [1925, 3256, 1924, 1925], [4441, 0, 0, 3701], [113, 4144, 148, 1], [105, 3034, 37, 783], [75, 74, 1591, 2295], [85, 3108, 148, 3209], [1, 2590, 0, 1], [2591, 2590, 2590, 2591], [4848, 2923, 4477, 1259], [1, 0, 2590, 1]]
 
GL(2,Integers(5180)).subgroup(gens)
 
Gens := [[1, 3256, 0, 1], [1, 0, 148, 1], [1, 28, 0, 1], [1037, 0, 0, 2073], [1925, 3256, 1924, 1925], [4441, 0, 0, 3701], [113, 4144, 148, 1], [105, 3034, 37, 783], [75, 74, 1591, 2295], [85, 3108, 148, 3209], [1, 2590, 0, 1], [2591, 2590, 2590, 2591], [4848, 2923, 4477, 1259], [1, 0, 2590, 1]];
 
sub<GL(2,Integers(5180))|Gens>;
 

The image H:=ρE(Gal(Q/Q))H:=\rho_E(\Gal(\overline{\Q}/\Q)) of the adelic Galois representation has level 5180=225737 5180 = 2^{2} \cdot 5 \cdot 7 \cdot 37 , index 27362736, genus 9797, and generators

(1325601),(101481),(12801),(1037002073),(1925325619241925),(4441003701),(11341441481),(105303437783),(757415912295),(8531081483209),(1259001),(2591259025902591),(4848292344771259),(1025901)\left(\begin{array}{rr} 1 & 3256 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 148 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 28 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1037 & 0 \\ 0 & 2073 \end{array}\right),\left(\begin{array}{rr} 1925 & 3256 \\ 1924 & 1925 \end{array}\right),\left(\begin{array}{rr} 4441 & 0 \\ 0 & 3701 \end{array}\right),\left(\begin{array}{rr} 113 & 4144 \\ 148 & 1 \end{array}\right),\left(\begin{array}{rr} 105 & 3034 \\ 37 & 783 \end{array}\right),\left(\begin{array}{rr} 75 & 74 \\ 1591 & 2295 \end{array}\right),\left(\begin{array}{rr} 85 & 3108 \\ 148 & 3209 \end{array}\right),\left(\begin{array}{rr} 1 & 2590 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2591 & 2590 \\ 2590 & 2591 \end{array}\right),\left(\begin{array}{rr} 4848 & 2923 \\ 4477 & 1259 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2590 & 1 \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[5180])K:=\Q(E[5180]) is a degree-6186958848061869588480 Galois extension of Q\Q with Gal(K/Q)\Gal(K/\Q) isomorphic to the projection of HH to GL2(Z/5180Z)\GL_2(\Z/5180\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 good 22 245=572 245 = 5 \cdot 7^{2}
55 additive 1010 49=72 49 = 7^{2}
77 additive 1414 25=52 25 = 5^{2}

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree dd for d=d= 37.
Its isogeny class 1225.b consists of 2 curves linked by isogenies of degree 37.

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 3.1.980.1 Z/2Z\Z/2\Z not in database
66 6.0.19208000.2 Z/2ZZ/2Z\Z/2\Z \oplus \Z/2\Z not in database
88 8.2.4020286921875.3 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 ord ord add add ss ord ord ord ord ord ord ord ord ord ss
λ\lambda-invariant(s) ? 1 - - 1,1 1 1 1 1 1 1 1 1 1 1,1
μ\mu-invariant(s) ? 0 - - 0,0 0 0 0 0 0 0 1 0 0 0,0

An entry ? indicates that the invariants have not yet been computed.

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

pp-adic regulators

pp-adic regulators are not yet computed for curves that are not Γ0\Gamma_0-optimal.