Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2+8867x-315488\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z+8867xz^2-315488z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+718200x-227836125\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(412, 8550)$ | $1.9569105828470900491096250240$ | $\infty$ |
| $(973/4, 32175/8)$ | $4.3738851491472089607164824147$ | $\infty$ |
| $(32, 0)$ | $0$ | $2$ |
Integral points
\( \left(32, 0\right) \), \((412,\pm 8550)\), \((4896,\pm 342608)\)
Invariants
| Conductor: | $N$ | = | \( 433200 \) | = | $2^{4} \cdot 3 \cdot 5^{2} \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $-86809218750000$ | = | $-1 \cdot 2^{4} \cdot 3^{4} \cdot 5^{10} \cdot 19^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{44957696}{50625} \) | = | $2^{17} \cdot 3^{-4} \cdot 5^{-4} \cdot 7^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $1.3610086812744993965763759769$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.41086907992080934219867125484$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0447680235871464$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.995908859215512$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $8.4359593405986509412402698862$ |
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| Real period: | $\Omega$ | ≈ | $0.32638043778209392318259374427$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 1\cdot2\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $11.013328410786128289240424372 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 11.013328411 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.326380 \cdot 8.435959 \cdot 16}{2^2} \\ & \approx 11.013328411\end{aligned}$$
Modular invariants
Modular form 433200.2.a.l
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1105920 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 2 curves in this isogeny class which might be optimal) | |
| Manin constant: | 1 (conditional*) |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II$ | additive | -1 | 4 | 4 | 0 |
| $3$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $5$ | $4$ | $I_{4}^{*}$ | additive | 1 | 2 | 10 | 4 |
| $19$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
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 | 4.12.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 760 = 2^{3} \cdot 5 \cdot 19 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 198 \\ 570 & 571 \end{array}\right),\left(\begin{array}{rr} 457 & 8 \\ 308 & 33 \end{array}\right),\left(\begin{array}{rr} 3 & 8 \\ 752 & 739 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 753 & 8 \\ 752 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 48 & 77 \end{array}\right),\left(\begin{array}{rr} 385 & 572 \\ 572 & 381 \end{array}\right),\left(\begin{array}{rr} 126 & 19 \\ 233 & 738 \end{array}\right)$.
The torsion field $K:=\Q(E[760])$ is a degree-$1891123200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/760\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 475 = 5^{2} \cdot 19 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 144400 = 2^{4} \cdot 5^{2} \cdot 19^{2} \) |
| $5$ | additive | $18$ | \( 17328 = 2^{4} \cdot 3 \cdot 19^{2} \) |
| $19$ | additive | $110$ | \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 433200l
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 21660n1, its twist by $-20$.
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.