Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-502514414x+20739075963348\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-502514414xz^2+20739075963348z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-8040230627x+1327292821423646\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-33492, 16746)$ | $0$ | $2$ |
Integral points
\( \left(-33492, 16746\right) \)
Invariants
| Conductor: | $N$ | = | \( 436810 \) | = | $2 \cdot 5 \cdot 11^{2} \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $-177683654509230261148844032000$ | = | $-1 \cdot 2^{32} \cdot 5^{3} \cdot 11^{7} \cdot 19^{8} $ |
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| j-invariant: | $j$ | = | \( -\frac{168380411424176601}{2131914391552000} \) | = | $-1 \cdot 2^{-32} \cdot 3^{3} \cdot 5^{-3} \cdot 11^{-1} \cdot 13^{3} \cdot 19^{-2} \cdot 14159^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.2942134361666958948313230120$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.6230463101842903927958375071$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0136269029301013$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.7632827751126765$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.027215000884121824756659254986$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ 2\cdot3\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.3063200424378475883196442393 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 1.306320042 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.027215 \cdot 1.000000 \cdot 48}{2^2} \\ & \approx 1.306320042\end{aligned}$$
Modular invariants
Modular form 436810.2.a.s
For more coefficients, see the Downloads section to the right.
| Modular degree: | 398131200 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 3 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{32}$ | nonsplit multiplicative | 1 | 1 | 32 | 32 |
| $5$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $11$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $19$ | $2$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 8360 = 2^{3} \cdot 5 \cdot 11 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 609 & 608 \\ 6118 & 8095 \end{array}\right),\left(\begin{array}{rr} 2699 & 2698 \\ 1938 & 3915 \end{array}\right),\left(\begin{array}{rr} 8353 & 8 \\ 8352 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 2639 & 0 \\ 0 & 8359 \end{array}\right),\left(\begin{array}{rr} 512 & 437 \\ 8075 & 5718 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 8354 & 8355 \end{array}\right),\left(\begin{array}{rr} 2908 & 2641 \\ 551 & 4846 \end{array}\right)$.
The torsion field $K:=\Q(E[8360])$ is a degree-$24962826240000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8360\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 218405 = 5 \cdot 11^{2} \cdot 19^{2} \) |
| $3$ | good | $2$ | \( 87362 = 2 \cdot 11^{2} \cdot 19^{2} \) |
| $5$ | split multiplicative | $6$ | \( 87362 = 2 \cdot 11^{2} \cdot 19^{2} \) |
| $11$ | additive | $72$ | \( 3610 = 2 \cdot 5 \cdot 19^{2} \) |
| $19$ | additive | $200$ | \( 1210 = 2 \cdot 5 \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 436810.s
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 2090.g4, its twist by $209$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.