Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-511839075x-4457026862750\)
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(homogenize, simplify) |
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\(y^2z=x^3-511839075xz^2-4457026862750z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-511839075x-4457026862750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-13090, 0)$ | $0$ | $2$ |
Integral points
\( \left(-13090, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 435600 \) | = | $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2}$ |
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| Discriminant: | $\Delta$ | = | $119304364532094720000000$ | = | $2^{14} \cdot 3^{14} \cdot 5^{7} \cdot 11^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{179415687049201}{1443420} \) | = | $2^{-2} \cdot 3^{-8} \cdot 5^{-1} \cdot 11^{-1} \cdot 56401^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.5996349677126665924237847849$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.35351505020243097797757858938$ |
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| $abc$ quality: | $Q$ | ≈ | $0.990076389601979$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.527689005451904$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.031740693021772364373254082277$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2^{2}\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.0157021766967156599441306329 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 1.015702177 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.031741 \cdot 1.000000 \cdot 32}{2^2} \\ & \approx 1.015702177\end{aligned}$$
Modular invariants
Modular form 435600.2.a.jr
For more coefficients, see the Downloads section to the right.
| Modular degree: | 70778880 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 (conditional*) |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}^{*}$ | additive | -1 | 4 | 14 | 2 |
| $3$ | $4$ | $I_{8}^{*}$ | additive | -1 | 2 | 14 | 8 |
| $5$ | $2$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $11$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
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 | 16.24.0.12 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2104 & 2639 \\ 977 & 2630 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 2542 & 2627 \end{array}\right),\left(\begin{array}{rr} 1759 & 2624 \\ 872 & 2511 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 2302 & 653 \\ 305 & 1298 \end{array}\right),\left(\begin{array}{rr} 1904 & 2635 \\ 1245 & 14 \end{array}\right),\left(\begin{array}{rr} 13 & 16 \\ 924 & 985 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 2636 & 2637 \end{array}\right),\left(\begin{array}{rr} 2625 & 16 \\ 2624 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[2640])$ is a degree-$38928384000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2640\Z)$.
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 | $2$ | \( 27225 = 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
| $3$ | additive | $8$ | \( 48400 = 2^{4} \cdot 5^{2} \cdot 11^{2} \) |
| $5$ | additive | $18$ | \( 17424 = 2^{4} \cdot 3^{2} \cdot 11^{2} \) |
| $11$ | additive | $72$ | \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 435600jr
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 330b4, its twist by $-660$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.