Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-1743195x-5684482550\)
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(homogenize, simplify) |
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\(y^2z=x^3-1743195xz^2-5684482550z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1743195x-5684482550\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(6335, 487350)$ | $2.1281486199442578721094560681$ | $\infty$ |
Integral points
\((6335,\pm 487350)\)
Invariants
| Conductor: | $N$ | = | \( 478800 \) | = | $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $-13620349484733616128000$ | = | $-1 \cdot 2^{15} \cdot 3^{12} \cdot 5^{3} \cdot 7 \cdot 19^{7} $ |
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| j-invariant: | $j$ | = | \( -\frac{1569510182075597}{36491419872936} \) | = | $-1 \cdot 2^{-3} \cdot 3^{-6} \cdot 7^{-1} \cdot 19^{-7} \cdot 251^{3} \cdot 463^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9278866345774520911736359646$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.2830738315749268424085913914$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0052902524628846$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.468612774739021$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.1281486199442578721094560681$ |
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| Real period: | $\Omega$ | ≈ | $0.054344060500402859953195669367$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 56 $ = $ 2\cdot2\cdot2\cdot1\cdot7 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.4765252919415777278447360549 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.476525292 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.054344 \cdot 2.128149 \cdot 56}{1^2} \\ & \approx 6.476525292\end{aligned}$$
Modular invariants
Modular form 478800.2.a.r
For more coefficients, see the Downloads section to the right.
| Modular degree: | 29417472 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{7}^{*}$ | additive | -1 | 4 | 15 | 3 |
| $3$ | $2$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
| $5$ | $2$ | $III$ | additive | -1 | 2 | 3 | 0 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $19$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
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 \( 5320 = 2^{3} \cdot 5 \cdot 7 \cdot 19 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 4257 & 2 \\ 4257 & 3 \end{array}\right),\left(\begin{array}{rr} 3991 & 2 \\ 3991 & 3 \end{array}\right),\left(\begin{array}{rr} 2661 & 2 \\ 2661 & 3 \end{array}\right),\left(\begin{array}{rr} 5319 & 2 \\ 5318 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 5319 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 3041 & 2 \\ 3041 & 3 \end{array}\right),\left(\begin{array}{rr} 4201 & 2 \\ 4201 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[5320])$ is a degree-$91500104908800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5320\Z)$.
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 | $4$ | \( 5985 = 3^{2} \cdot 5 \cdot 7 \cdot 19 \) |
| $3$ | additive | $6$ | \( 53200 = 2^{4} \cdot 5^{2} \cdot 7 \cdot 19 \) |
| $5$ | additive | $10$ | \( 19152 = 2^{4} \cdot 3^{2} \cdot 7 \cdot 19 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \) |
| $19$ | split multiplicative | $20$ | \( 25200 = 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 478800.r consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 19950.e1, its twist by $12$.
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.