Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-5110140x-4459266335\)
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(homogenize, simplify) |
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\(y^2z=x^3-5110140xz^2-4459266335z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-5110140x-4459266335\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 476100 \) | = | $2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 23^{2}$ |
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| Discriminant: | $\Delta$ | = | $-49941201994463905200$ | = | $-1 \cdot 2^{4} \cdot 3^{13} \cdot 5^{2} \cdot 23^{8} $ |
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| j-invariant: | $j$ | = | \( -\frac{646266880}{2187} \) | = | $-1 \cdot 2^{14} \cdot 3^{-7} \cdot 5 \cdot 7^{3} \cdot 23$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.6439419536429815861541414458$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.49498238023617155232052032323$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9748773066603298$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.433408311637468$ | |||
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: | $\Omega$ | ≈ | $0.050196491980534087379231014234$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 18 $ = $ 3\cdot2\cdot1\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.90353685564961357282615825621 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.903536856 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.050196 \cdot 1.000000 \cdot 18}{1^2} \\ & \approx 0.903536856\end{aligned}$$
Modular invariants
Modular form 476100.2.a.b
For more coefficients, see the Downloads section to the right.
| Modular degree: | 21143808 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $3$ | $2$ | $I_{7}^{*}$ | additive | -1 | 2 | 13 | 7 |
| $5$ | $1$ | $II$ | additive | 1 | 2 | 2 | 0 |
| $23$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
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 label 6.2.0.a.1, level \( 6 = 2 \cdot 3 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 5 & 2 \\ 5 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 5 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 2 \\ 4 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[6])$ is a degree-$144$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6\Z)$.
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 | $2$ | \( 119025 = 3^{2} \cdot 5^{2} \cdot 23^{2} \) |
| $3$ | additive | $8$ | \( 52900 = 2^{2} \cdot 5^{2} \cdot 23^{2} \) |
| $5$ | additive | $10$ | \( 19044 = 2^{2} \cdot 3^{2} \cdot 23^{2} \) |
| $23$ | additive | $200$ | \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 476100b consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 158700r1, its twist by $69$.
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 $0$.