Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-109875x+4418750\)
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(homogenize, simplify) |
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\(y^2z=x^3-109875xz^2+4418750z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-109875x+4418750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-334, 1964)$ | $6.2160148741107178822870753131$ | $\infty$ |
| $(-350, 0)$ | $0$ | $2$ |
Integral points
\( \left(-350, 0\right) \), \((-334,\pm 1964)\)
Invariants
| Conductor: | $N$ | = | \( 412200 \) | = | $2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 229$ |
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| Discriminant: | $\Delta$ | = | $76458978000000000$ | = | $2^{10} \cdot 3^{6} \cdot 5^{9} \cdot 229^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{100615028}{52441} \) | = | $2^{2} \cdot 229^{-2} \cdot 293^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9314840507269448314300040996$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.40252317839930638639921478666$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8085224261522714$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.591457265197656$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $6.2160148741107178822870753131$ |
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| Real period: | $\Omega$ | ≈ | $0.30256033940282632678655905345$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.5228782801758222599471015175 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.522878280 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.302560 \cdot 6.216015 \cdot 16}{2^2} \\ & \approx 7.522878280\end{aligned}$$
Modular invariants
Modular form 412200.2.a.q
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2488320 |
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| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III^{*}$ | additive | -1 | 3 | 10 | 0 |
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 0 |
| $229$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4580 = 2^{2} \cdot 5 \cdot 229 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 4577 & 4 \\ 4576 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3437 & 1146 \\ 1144 & 3435 \end{array}\right),\left(\begin{array}{rr} 3668 & 1 \\ 3663 & 0 \end{array}\right),\left(\begin{array}{rr} 3441 & 4 \\ 2302 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[4580])$ is a degree-$10513909555200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4580\Z)$.
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 | $2$ | \( 45 = 3^{2} \cdot 5 \) |
| $3$ | additive | $6$ | \( 45800 = 2^{3} \cdot 5^{2} \cdot 229 \) |
| $5$ | additive | $14$ | \( 16488 = 2^{3} \cdot 3^{2} \cdot 229 \) |
| $229$ | split multiplicative | $230$ | \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 412200.q
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 45800.b1, 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.