Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2+115632x-10973248\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z+115632xz^2-10973248z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+9366165x-7971399270\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(231101/361, 123650982/6859)$ | $12.624972383184195459717006824$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 400400 \) | = | $2^{4} \cdot 5^{2} \cdot 7 \cdot 11 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-150602023488716800$ | = | $-1 \cdot 2^{23} \cdot 5^{2} \cdot 7^{3} \cdot 11^{5} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( \frac{1669760225634695}{1470722885632} \) | = | $2^{-11} \cdot 5 \cdot 7^{-3} \cdot 11^{-5} \cdot 13^{-1} \cdot 69379^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9835824817734227492991190008$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0221956491411273774484269905$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9175720200265711$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.6114190965520367$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $12.624972383184195459717006824$ |
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| Real period: | $\Omega$ | ≈ | $0.17882706503264830327829122375$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 6 $ = $ 2\cdot1\cdot3\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $13.546120544418413734162600761 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 13.546120544 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.178827 \cdot 12.624972 \cdot 6}{1^2} \\ & \approx 13.546120544\end{aligned}$$
Modular invariants
Modular form 400400.2.a.fg
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3801600 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{15}^{*}$ | additive | -1 | 4 | 23 | 11 |
| $5$ | $1$ | $II$ | additive | 1 | 2 | 2 | 0 |
| $7$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $11$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
| $13$ | $1$ | $I_{1}$ | nonsplit 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 \( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 4005 & 2 \\ 4005 & 3 \end{array}\right),\left(\begin{array}{rr} 6007 & 2 \\ 6007 & 3 \end{array}\right),\left(\begin{array}{rr} 4369 & 2 \\ 4369 & 3 \end{array}\right),\left(\begin{array}{rr} 3433 & 2 \\ 3433 & 3 \end{array}\right),\left(\begin{array}{rr} 8007 & 2 \\ 8006 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 8007 & 0 \end{array}\right),\left(\begin{array}{rr} 4929 & 2 \\ 4929 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[8008])$ is a degree-$535623421132800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8008\Z)$.
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 | $4$ | \( 25025 = 5^{2} \cdot 7 \cdot 11 \cdot 13 \) |
| $3$ | good | $2$ | \( 57200 = 2^{4} \cdot 5^{2} \cdot 11 \cdot 13 \) |
| $5$ | additive | $10$ | \( 1456 = 2^{4} \cdot 7 \cdot 13 \) |
| $7$ | split multiplicative | $8$ | \( 57200 = 2^{4} \cdot 5^{2} \cdot 11 \cdot 13 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 36400 = 2^{4} \cdot 5^{2} \cdot 7 \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 30800 = 2^{4} \cdot 5^{2} \cdot 7 \cdot 11 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 400400fg consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 50050bo1, 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.