Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+69000x+6612500\)
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(homogenize, simplify) |
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\(y^2z=x^3+69000xz^2+6612500z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+69000x+6612500\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-575/16, 129375/64)$ | $3.4394238729477794425808553560$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 476100 \) | = | $2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 23^{2}$ |
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| Discriminant: | $\Delta$ | = | $-39913843500000000$ | = | $-1 \cdot 2^{8} \cdot 3^{8} \cdot 5^{9} \cdot 23^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{8192}{9} \) | = | $2^{13} \cdot 3^{-2}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.8720030537548941266254815680$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.1303531992603202956692201726$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8403427576978969$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.4451058085758763$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.4394238729477794425808553560$ |
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| Real period: | $\Omega$ | ≈ | $0.24121892411514127324537512768$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 1\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.6372330096671656743233629720 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.637233010 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.241219 \cdot 3.439424 \cdot 8}{1^2} \\ & \approx 6.637233010\end{aligned}$$
Modular invariants
Modular form 476100.2.a.v
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2580480 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $3$ | $2$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $5$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 0 |
| $23$ | $2$ | $III$ | additive | -1 | 2 | 3 | 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 \( 230 = 2 \cdot 5 \cdot 23 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 229 & 2 \\ 228 & 3 \end{array}\right),\left(\begin{array}{rr} 51 & 2 \\ 51 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 229 & 0 \end{array}\right),\left(\begin{array}{rr} 47 & 2 \\ 47 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[230])$ is a degree-$384721920$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/230\Z)$.
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 | $2$ | \( 1035 = 3^{2} \cdot 5 \cdot 23 \) |
| $3$ | additive | $8$ | \( 52900 = 2^{2} \cdot 5^{2} \cdot 23^{2} \) |
| $5$ | additive | $14$ | \( 19044 = 2^{2} \cdot 3^{2} \cdot 23^{2} \) |
| $23$ | additive | $156$ | \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 476100.v consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 158700.bd1, its twist by $-15$.
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.