Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+2125x+131250\)
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(homogenize, simplify) |
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\(y^2z=x^3+2125xz^2+131250z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+2125x+131250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-25, 250)$ | $1.2635999696240767340151167314$ | $\infty$ |
| $(25/9, 10000/27)$ | $2.8019405284488912308775125211$ | $\infty$ |
Integral points
\((-25,\pm 250)\), \((41,\pm 536)\), \((114,\pm 1362)\), \((521,\pm 11944)\)
Invariants
| Conductor: | $N$ | = | \( 402800 \) | = | $2^{4} \cdot 5^{2} \cdot 19 \cdot 53$ |
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| Discriminant: | $\Delta$ | = | $-8056000000000$ | = | $-1 \cdot 2^{12} \cdot 5^{9} \cdot 19 \cdot 53 $ |
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| j-invariant: | $j$ | = | \( \frac{132651}{1007} \) | = | $3^{3} \cdot 17^{3} \cdot 19^{-1} \cdot 53^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.1579690914163866247855640438$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.74225652346913396558223757758$ |
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| $abc$ quality: | $Q$ | ≈ | $0.7335923153258521$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.874035502674394$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.3514734955111556702712060258$ |
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| Real period: | $\Omega$ | ≈ | $0.53802607241385392019125056546$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2^{2}\cdot2\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $14.425440972711977295927276254 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 14.425440973 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.538026 \cdot 3.351473 \cdot 8}{1^2} \\ & \approx 14.425440973\end{aligned}$$
Modular invariants
Modular form 402800.2.a.q
For more coefficients, see the Downloads section to the right.
| Modular degree: | 389120 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}^{*}$ | additive | -1 | 4 | 12 | 0 |
| $5$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 0 |
| $19$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $53$ | $1$ | $I_{1}$ | 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 \( 10070 = 2 \cdot 5 \cdot 19 \cdot 53 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 10069 & 2 \\ 10068 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 10069 & 0 \end{array}\right),\left(\begin{array}{rr} 2281 & 2 \\ 2281 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 8057 & 2 \\ 8057 & 3 \end{array}\right),\left(\begin{array}{rr} 4771 & 2 \\ 4771 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[10070])$ is a degree-$1372042030694400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10070\Z)$.
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 | $2$ | \( 5035 = 5 \cdot 19 \cdot 53 \) |
| $5$ | additive | $14$ | \( 16112 = 2^{4} \cdot 19 \cdot 53 \) |
| $19$ | split multiplicative | $20$ | \( 21200 = 2^{4} \cdot 5^{2} \cdot 53 \) |
| $53$ | nonsplit multiplicative | $54$ | \( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 402800.q consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 25175.c1, its twist by $-20$.
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.