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Properties

Label 471510.e2
Conductor $471510$
Discriminant $1.089\times 10^{17}$
j-invariant \( \frac{53540005609}{30950400} \)
CM no
Rank $2$
Torsion structure \(\Z/{2}\Z\)

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Show commands: Magma / Oscar / PariGP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy=x^3-x^2-119430x-491724\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz=x^3-x^2z-119430xz^2-491724z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-1910883x-33381218\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([1, -1, 0, -119430, -491724])
 
gp: E = ellinit([1, -1, 0, -119430, -491724])
 
magma: E := EllipticCurve([1, -1, 0, -119430, -491724]);
 
oscar: E = EllipticCurve([1, -1, 0, -119430, -491724])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

\(\Z \oplus \Z \oplus \Z/{2}\Z\)

magma: MordellWeilGroup(E);
 

Infinite order Mordell-Weil generators and heights

$P$ =  \(\left(-68, 2738\right)\) Copy content Toggle raw display \(\left(-159, 3882\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $0.98404564364111557238977617648$$2.0820359286890351333162285054$

sage: E.gens()
 
magma: Generators(E);
 
gp: E.gen
 

Torsion generators

\( \left(348, -174\right) \) Copy content Toggle raw display

comment: Torsion subgroup
 
sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 
oscar: torsion_structure(E)
 

Integral points

\( \left(-237, 3921\right) \), \( \left(-237, -3684\right) \), \( \left(-159, 3882\right) \), \( \left(-159, -3723\right) \), \( \left(-132, 3666\right) \), \( \left(-132, -3534\right) \), \( \left(-68, 2738\right) \), \( \left(-68, -2670\right) \), \( \left(-15, 1146\right) \), \( \left(-15, -1131\right) \), \( \left(348, -174\right) \), \( \left(517, 8445\right) \), \( \left(517, -8962\right) \), \( \left(1788, 73266\right) \), \( \left(1788, -75054\right) \), \( \left(5340, 386706\right) \), \( \left(5340, -392046\right) \), \( \left(9903, 979896\right) \), \( \left(9903, -989799\right) \) Copy content Toggle raw display

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

Invariants

Conductor: \( 471510 \)  =  $2 \cdot 3^{2} \cdot 5 \cdot 13^{2} \cdot 31$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $108906526900454400 $  =  $2^{10} \cdot 3^{7} \cdot 5^{2} \cdot 13^{7} \cdot 31 $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: \( \frac{53540005609}{30950400} \)  =  $2^{-10} \cdot 3^{-1} \cdot 5^{-2} \cdot 13^{-1} \cdot 31^{-1} \cdot 3769^{3}$
comment: j-invariant
 
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
oscar: j_invariant(E)
 
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $1.9585969698574375903520018198\dots$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $0.12681614679261437662763548056\dots$
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
$abc$ quality: $0.9297516717491048\dots$
Szpiro ratio: $3.573649166568941\dots$

BSD invariants

Analytic rank: $2$
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Regulator: $1.9020375295302496423389499985\dots$
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: $0.28089283485590839215903009486\dots$
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: $ 64 $  = $ 2\cdot2^{2}\cdot2\cdot2^{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: $2$
comment: Torsion order
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
oscar: prod(torsion_structure(E)[1])
 
Analytic order of Ш: $1$ ( rounded)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Special value: $ L^{(2)}(E,1)/2! $ ≈ $ 8.5482994187532863118470999864 $
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);
 

BSD formula

$\displaystyle 8.548299419 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.280893 \cdot 1.902038 \cdot 64}{2^2} \approx 8.548299419$

# 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 analyiic 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 471510.2.a.e

\( q - q^{2} + q^{4} - q^{5} - 2 q^{7} - q^{8} + q^{10} - 6 q^{11} + 2 q^{14} + q^{16} - 6 q^{17} + 6 q^{19} + O(q^{20}) \) 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: 6021120
comment: Modular degree
 
sage: E.modular_degree()
 
gp: ellmoddegree(E)
 
magma: ModularDegree(E);
 
$ \Gamma_0(N) $-optimal: yes
Manin constant: 1
comment: Manin constant
 
magma: ManinConstant(E);
 

Local data

This elliptic curve is not semistable. There are 5 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $2$ $I_{10}$ Non-split multiplicative 1 1 10 10
$3$ $4$ $I_{1}^{*}$ Additive -1 2 7 1
$5$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$13$ $4$ $I_{1}^{*}$ Additive 1 2 7 1
$31$ $1$ $I_{1}$ Non-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 $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 2.3.0.1

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 = [[18137, 30226, 30224, 18135], [1, 2, 2, 5], [29642, 1, 9359, 0], [1, 4, 0, 1], [48357, 4, 48356, 5], [3722, 1, 22319, 0], [24181, 4, 2, 9], [1, 0, 4, 1], [3, 4, 8, 11], [32242, 1, 32239, 0], [29017, 4, 9674, 9]]
 
GL(2,Integers(48360)).subgroup(gens)
 
Gens := [[18137, 30226, 30224, 18135], [1, 2, 2, 5], [29642, 1, 9359, 0], [1, 4, 0, 1], [48357, 4, 48356, 5], [3722, 1, 22319, 0], [24181, 4, 2, 9], [1, 0, 4, 1], [3, 4, 8, 11], [32242, 1, 32239, 0], [29017, 4, 9674, 9]];
 
sub<GL(2,Integers(48360))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 48360 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \cdot 31 \), index $12$, genus $0$, and generators

$\left(\begin{array}{rr} 18137 & 30226 \\ 30224 & 18135 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 29642 & 1 \\ 9359 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 48357 & 4 \\ 48356 & 5 \end{array}\right),\left(\begin{array}{rr} 3722 & 1 \\ 22319 & 0 \end{array}\right),\left(\begin{array}{rr} 24181 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 32242 & 1 \\ 32239 & 0 \end{array}\right),\left(\begin{array}{rr} 29017 & 4 \\ 9674 & 9 \end{array}\right)$.

Input positive integer $m$ to see the generators of the reduction of $H$ to $\mathrm{GL}_2(\Z/m\Z)$:

The torsion field $K:=\Q(E[48360])$ is a degree-$69004991397888000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/48360\Z)$.

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 471510.e consists of 2 curves linked by isogenies of degree 2.

Twists

The minimal quadratic twist of this elliptic curve is 12090.n2, its twist by $-39$.

Iwasawa invariants

No Iwasawa invariant data is available for this curve.

$p$-adic regulators

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.