Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-x^2-40860x+2144016\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-x^2z-40860xz^2+2144016z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-653763x+136563262\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(48, 516\right)\) | \(\left(183, 786\right)\) |
$\hat{h}(P)$ | ≈ | $1.4404878512388360165426717315$ | $2.6893539696541714295784955636$ |
Torsion generators
\( \left(168, -84\right) \)
Integral points
\( \left(-224, 252\right) \), \( \left(-224, -28\right) \), \( \left(-195, 1731\right) \), \( \left(-195, -1536\right) \), \( \left(-120, 2364\right) \), \( \left(-120, -2244\right) \), \( \left(48, 516\right) \), \( \left(48, -564\right) \), \( \left(168, -84\right) \), \( \left(183, 786\right) \), \( \left(183, -969\right) \), \( \left(897, 25755\right) \), \( \left(897, -26652\right) \), \( \left(3072, 168348\right) \), \( \left(3072, -171420\right) \)
Invariants
Conductor: | \( 468270 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 11^{2} \cdot 43$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $2399030899214400 $ | = | $2^{6} \cdot 3^{9} \cdot 5^{2} \cdot 11^{6} \cdot 43 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{5841725401}{1857600} \) | = | $2^{-6} \cdot 3^{-3} \cdot 5^{-2} \cdot 43^{-1} \cdot 1801^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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Endomorphism ring: | $\Z$ | |||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $1.6549858861567405936500263321\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.093267894576499524078568075343\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0158843938866033\dots$ | |||
Szpiro ratio: | $3.329095666651105\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $3.7730487127838480728066714189\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.42434376503208338584607681319\dots$ | 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);
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Tamagawa product: | $ 32 $ = $ 2\cdot2^{2}\cdot2\cdot2\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)
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Torsion order: | $2$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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Analytic order of Ш: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L^{(2)}(E,1)/2! $ ≈ $ 12.808557571457231200568517740 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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BSD formula
$\displaystyle 12.808557571 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.424344 \cdot 3.773049 \cdot 32}{2^2} \approx 12.808557571$
Modular invariants
Modular form 468270.2.a.v
For more coefficients, see the Downloads section to the right.
Modular degree: | 3225600 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{6}$ | Non-split multiplicative | 1 | 1 | 6 | 6 |
$3$ | $4$ | $I_{3}^{*}$ | Additive | -1 | 2 | 9 | 3 |
$5$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$11$ | $2$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$43$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
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 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 5160 = 2^{3} \cdot 3 \cdot 5 \cdot 43 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3097 & 4 \\ 1034 & 9 \end{array}\right),\left(\begin{array}{rr} 3442 & 1 \\ 3439 & 0 \end{array}\right),\left(\begin{array}{rr} 1937 & 3226 \\ 3224 & 1935 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3962 & 1 \\ 599 & 0 \end{array}\right),\left(\begin{array}{rr} 2581 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 5157 & 4 \\ 5156 & 5 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right)$.
The torsion field $K:=\Q(E[5160])$ is a degree-$9842652610560$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5160\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 468270.v
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 1290.g2, its twist by $33$.
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.