Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-116803200x+485943896500\)
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(homogenize, simplify) |
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\(y^2z=x^3-116803200xz^2+485943896500z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-116803200x+485943896500\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(34385/4, 2737575/8)$ | $2.5276977616494496594750187630$ | $\infty$ |
| $(23805, 3345925)$ | $3.2666115650974711920530617077$ | $\infty$ |
Integral points
\((6605,\pm 51075)\), \((23805,\pm 3345925)\), \((26405,\pm 3976425)\)
Invariants
| Conductor: | $N$ | = | \( 476100 \) | = | $2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 23^{2}$ |
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| Discriminant: | $\Delta$ | = | $-26260805804130540000000$ | = | $-1 \cdot 2^{8} \cdot 3^{6} \cdot 5^{7} \cdot 23^{9} $ |
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| j-invariant: | $j$ | = | \( -\frac{33554432}{5} \) | = | $-1 \cdot 2^{25} \cdot 5^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.3154246188034650860341225601$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.85231926406779908801376576314$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1398724641416618$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.151066297222018$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $7.9011044558872619611167782518$ |
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| Real period: | $\Omega$ | ≈ | $0.11489120124463309308983643416$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 1\cdot2\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $14.524278113539370706942591323 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 14.524278114 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.114891 \cdot 7.901104 \cdot 16}{1^2} \\ & \approx 14.524278114\end{aligned}$$
Modular invariants
Modular form 476100.2.a.u
For more coefficients, see the Downloads section to the right.
| Modular degree: | 57231360 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $23$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 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 level \( 230 = 2 \cdot 5 \cdot 23 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 229 & 2 \\ 228 & 3 \end{array}\right),\left(\begin{array}{rr} 51 & 2 \\ 51 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 229 & 0 \end{array}\right),\left(\begin{array}{rr} 47 & 2 \\ 47 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[230])$ is a degree-$384721920$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/230\Z)$.
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 | $2$ | \( 5175 = 3^{2} \cdot 5^{2} \cdot 23 \) |
| $3$ | additive | $6$ | \( 52900 = 2^{2} \cdot 5^{2} \cdot 23^{2} \) |
| $5$ | additive | $18$ | \( 19044 = 2^{2} \cdot 3^{2} \cdot 23^{2} \) |
| $23$ | additive | $156$ | \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 476100.u consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 10580.k1, its twist by $345$.
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.