Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-415298017x-13106707522729\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-415298017xz^2-13106707522729z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-538226230707x-611498472786986994\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(13097317/81, 46888636456/729)$ | $4.1854096402810982463505140158$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 436810 \) | = | $2 \cdot 5 \cdot 11^{2} \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $-69625340642576372981432806250$ | = | $-1 \cdot 2 \cdot 5^{5} \cdot 11^{13} \cdot 19^{9} $ |
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| j-invariant: | $j$ | = | \( -\frac{13856857998859}{121794818750} \) | = | $-1 \cdot 2^{-1} \cdot 5^{-5} \cdot 11^{-7} \cdot 24019^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.2169855741797262501773163468$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.80970870340571063313957398390$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9753285092561745$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.692613073289055$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.1854096402810982463505140158$ |
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| Real period: | $\Omega$ | ≈ | $0.014656891735363271460607722525$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 40 $ = $ 1\cdot5\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.4538038386298316734769759951 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.453803839 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.014657 \cdot 4.185410 \cdot 40}{1^2} \\ & \approx 2.453803839\end{aligned}$$
Modular invariants
Modular form 436810.2.a.k
For more coefficients, see the Downloads section to the right.
| Modular degree: | 510720000 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $11$ | $4$ | $I_{7}^{*}$ | additive | -1 | 2 | 13 | 7 |
| $19$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
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 $2$, genus $0$, and generators
$\left(\begin{array}{rr} 8359 & 2 \\ 8358 & 3 \end{array}\right),\left(\begin{array}{rr} 5721 & 2 \\ 5721 & 3 \end{array}\right),\left(\begin{array}{rr} 6271 & 2 \\ 6271 & 3 \end{array}\right),\left(\begin{array}{rr} 5017 & 2 \\ 5017 & 3 \end{array}\right),\left(\begin{array}{rr} 761 & 2 \\ 761 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4181 & 2 \\ 4181 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 8359 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[8360])$ is a degree-$599107829760000$ 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 | $4$ | \( 11495 = 5 \cdot 11^{2} \cdot 19 \) |
| $5$ | split multiplicative | $6$ | \( 87362 = 2 \cdot 11^{2} \cdot 19^{2} \) |
| $11$ | additive | $72$ | \( 3610 = 2 \cdot 5 \cdot 19^{2} \) |
| $19$ | additive | $110$ | \( 1210 = 2 \cdot 5 \cdot 11^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 436810.k consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 39710.k1, its twist by $209$.
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.