Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-286275x-47330750\)
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(homogenize, simplify) |
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\(y^2z=x^3-286275xz^2-47330750z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-286275x-47330750\)
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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 |
|---|---|---|
| $(-265, 3150)$ | $1.4444401338387610761085657789$ | $\infty$ |
| $(-190, 450)$ | $3.3409482553263054831150627933$ | $\infty$ |
| $(-415, 0)$ | $0$ | $2$ |
Integral points
\( \left(-415, 0\right) \), \((-265,\pm 3150)\), \((-190,\pm 450)\), \((609,\pm 2048)\), \((674,\pm 8118)\), \((1121,\pm 32256)\), \((1751,\pm 69426)\), \((2759,\pm 142002)\)
Invariants
| Conductor: | $N$ | = | \( 478800 \) | = | $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $533750611968000000$ | = | $2^{24} \cdot 3^{7} \cdot 5^{6} \cdot 7^{2} \cdot 19 $ |
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| j-invariant: | $j$ | = | \( \frac{55611739513}{11440128} \) | = | $2^{-12} \cdot 3^{-1} \cdot 7^{-2} \cdot 11^{3} \cdot 19^{-1} \cdot 347^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1168998742713839348011252702$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.069727593160333592385890863668$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9136325031445056$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.7699811392108815$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.8157778241627990247955508038$ |
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| Real period: | $\Omega$ | ≈ | $0.20938312766973412498419779901$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2^{2}\cdot2^{2}\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $16.133481967772060225091506181 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 16.133481968 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.209383 \cdot 4.815778 \cdot 64}{2^2} \\ & \approx 16.133481968\end{aligned}$$
Modular invariants
Modular form 478800.2.a.dq
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4718592 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{16}^{*}$ | additive | -1 | 4 | 24 | 12 |
| $3$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $7$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $19$ | $1$ | $I_{1}$ | 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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1079 & 1080 \\ 2210 & 1529 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 1756 & 1825 \\ 1775 & 6 \end{array}\right),\left(\begin{array}{rr} 1091 & 630 \\ 290 & 971 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 2274 & 2275 \end{array}\right),\left(\begin{array}{rr} 2273 & 8 \\ 2272 & 9 \end{array}\right),\left(\begin{array}{rr} 1823 & 0 \\ 0 & 2279 \end{array}\right),\left(\begin{array}{rr} 1664 & 1365 \\ 755 & 454 \end{array}\right)$.
The torsion field $K:=\Q(E[2280])$ is a degree-$90773913600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2280\Z)$.
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 | $2$ | \( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \) |
| $3$ | additive | $8$ | \( 53200 = 2^{4} \cdot 5^{2} \cdot 7 \cdot 19 \) |
| $5$ | additive | $14$ | \( 19152 = 2^{4} \cdot 3^{2} \cdot 7 \cdot 19 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 68400 = 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 19 \) |
| $19$ | split multiplicative | $20$ | \( 25200 = 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 478800dq
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 798b1, its twist by $60$.
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.