Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-75755x+8025370\)
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(homogenize, simplify) |
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\(y^2z=x^3-75755xz^2+8025370z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-75755x+8025370\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(159, 2)$ | $0.79760117021222963603135951659$ | $\infty$ |
| $(-5409/25, 472486/125)$ | $8.4924537946287965045908862747$ | $\infty$ |
Integral points
\((159,\pm 2)\), \((166,\pm 156)\)
Invariants
| Conductor: | $N$ | = | \( 402800 \) | = | $2^{4} \cdot 5^{2} \cdot 19 \cdot 53$ |
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| Discriminant: | $\Delta$ | = | $412467200$ | = | $2^{14} \cdot 5^{2} \cdot 19 \cdot 53 $ |
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| j-invariant: | $j$ | = | \( \frac{469523855873385}{4028} \) | = | $2^{-2} \cdot 3^{3} \cdot 5 \cdot 19^{-1} \cdot 53^{-1} \cdot 109^{3} \cdot 139^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2386960835910106296547474294$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.27730925095871525780405541907$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9090451083892759$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.511443974903464$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $6.7139394904529493424632181998$ |
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| Real period: | $\Omega$ | ≈ | $1.1670662616027144391729781242$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 2\cdot1\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $15.671224523499514120321232945 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 15.671224523 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.167066 \cdot 6.713939 \cdot 2}{1^2} \\ & \approx 15.671224523\end{aligned}$$
Modular invariants
Modular form 402800.2.a.w
For more coefficients, see the Downloads section to the right.
| Modular degree: | 529920 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}^{*}$ | additive | -1 | 4 | 14 | 2 |
| $5$ | $1$ | $II$ | additive | 1 | 2 | 2 | 0 |
| $19$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $53$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
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 \( 4028 = 2^{2} \cdot 19 \cdot 53 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 2757 & 2 \\ 2757 & 3 \end{array}\right),\left(\begin{array}{rr} 4027 & 2 \\ 4026 & 3 \end{array}\right),\left(\begin{array}{rr} 2281 & 2 \\ 2281 & 3 \end{array}\right),\left(\begin{array}{rr} 2015 & 2 \\ 2015 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 4027 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[4028])$ is a degree-$45734734356480$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4028\Z)$.
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 | $2$ | \( 25175 = 5^{2} \cdot 19 \cdot 53 \) |
| $5$ | additive | $10$ | \( 16112 = 2^{4} \cdot 19 \cdot 53 \) |
| $19$ | split multiplicative | $20$ | \( 21200 = 2^{4} \cdot 5^{2} \cdot 53 \) |
| $53$ | split multiplicative | $54$ | \( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 402800w consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 50350i1, its twist by $-4$.
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.