Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-4062492x-12123426924\)
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(homogenize, simplify) |
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\(y^2z=x^3-4062492xz^2-12123426924z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-4062492x-12123426924\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(3642, 146250\right) \) | $2.5145527543393879669233641437$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([3642:146250:1]\) | $2.5145527543393879669233641437$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(3642, 146250\right) \) | $2.5145527543393879669233641437$ | $\infty$ |
Integral points
\((3642,\pm 146250)\)
\([3642:\pm 146250:1]\)
\((3642,\pm 146250)\)
Invariants
| Conductor: | $N$ | = | \( 405720 \) | = | $2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 23$ |
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| Minimal Discriminant: | $\Delta$ | = | $-59203281249787500000000$ | = | $-1 \cdot 2^{8} \cdot 3^{6} \cdot 5^{11} \cdot 7^{10} \cdot 23 $ |
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| j-invariant: | $j$ | = | \( -\frac{140654416896}{1123046875} \) | = | $-1 \cdot 2^{10} \cdot 3^{3} \cdot 5^{-11} \cdot 7^{2} \cdot 23^{-1} \cdot 47^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.0524420078339808374752774939$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.41944595224720136457837284160$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9891109428942693$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.643228776380227$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.5145527543393879669233641437$ |
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| Real period: | $\Omega$ | ≈ | $0.046851061310635341084114426652$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 88 $ = $ 2^{2}\cdot2\cdot11\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $10.367232943089583691610276709 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 10.367232943 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.046851 \cdot 2.514553 \cdot 88}{1^2} \\ & \approx 10.367232943\end{aligned}$$
Modular invariants
Modular form 405720.2.a.gs
For more coefficients, see the Downloads section to the right.
| Modular degree: | 28858368 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{1}^{*}$ | additive | 1 | 3 | 8 | 0 |
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $11$ | $I_{11}$ | split multiplicative | -1 | 1 | 11 | 11 |
| $7$ | $1$ | $II^{*}$ | additive | -1 | 2 | 10 | 0 |
| $23$ | $1$ | $I_{1}$ | 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 \( 230 = 2 \cdot 5 \cdot 23 \), 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 | $2$ | \( 50715 = 3^{2} \cdot 5 \cdot 7^{2} \cdot 23 \) |
| $3$ | additive | $6$ | \( 45080 = 2^{3} \cdot 5 \cdot 7^{2} \cdot 23 \) |
| $5$ | split multiplicative | $6$ | \( 81144 = 2^{3} \cdot 3^{2} \cdot 7^{2} \cdot 23 \) |
| $7$ | additive | $20$ | \( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \) |
| $11$ | good | $2$ | \( 81144 = 2^{3} \cdot 3^{2} \cdot 7^{2} \cdot 23 \) |
| $23$ | split multiplicative | $24$ | \( 17640 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 405720.gs consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 45080.q1, its twist by $21$.
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.