Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-5643765x-4451538195\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-5643765xz^2-4451538195z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-90300243x-284988744722\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(41079/4, 8041047/8)$ | $9.0975858064546708404165021958$ | $\infty$ |
| $(10803/4, -10803/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 468270 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 11^{2} \cdot 43$ |
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| Discriminant: | $\Delta$ | = | $2938981643353323893160$ | = | $2^{3} \cdot 3^{8} \cdot 5 \cdot 11^{6} \cdot 43^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{15393836938735081}{2275690697640} \) | = | $2^{-3} \cdot 3^{-2} \cdot 5^{-1} \cdot 17^{3} \cdot 43^{-6} \cdot 14633^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.8440278913859069473503921355$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0957741106526668296217977281$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9789160041460154$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.461413785517756$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $9.0975858064546708404165021958$ |
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| Real period: | $\Omega$ | ≈ | $0.098919299195052789299746408206$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 1\cdot2^{2}\cdot1\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.5997072493654208090100307343 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.599707249 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.098919 \cdot 9.097586 \cdot 16}{2^2} \\ & \approx 3.599707249\end{aligned}$$
Modular invariants
Modular form 468270.2.a.d
For more coefficients, see the Downloads section to the right.
| Modular degree: | 24883200 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $3$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $11$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $43$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
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 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 56760 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 43 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10330 & 41283 \\ 54153 & 46432 \end{array}\right),\left(\begin{array}{rr} 22155 & 43648 \\ 37598 & 48797 \end{array}\right),\left(\begin{array}{rr} 30361 & 51612 \\ 37686 & 25873 \end{array}\right),\left(\begin{array}{rr} 10330 & 41283 \\ 14421 & 46432 \end{array}\right),\left(\begin{array}{rr} 18919 & 5148 \\ 21494 & 30887 \end{array}\right),\left(\begin{array}{rr} 51599 & 0 \\ 0 & 56759 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 56710 & 56751 \end{array}\right),\left(\begin{array}{rr} 56749 & 12 \\ 56748 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[56760])$ is a degree-$16240376807424000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/56760\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \) |
| $3$ | additive | $8$ | \( 605 = 5 \cdot 11^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 93654 = 2 \cdot 3^{2} \cdot 11^{2} \cdot 43 \) |
| $11$ | additive | $62$ | \( 3870 = 2 \cdot 3^{2} \cdot 5 \cdot 43 \) |
| $43$ | nonsplit multiplicative | $44$ | \( 10890 = 2 \cdot 3^{2} \cdot 5 \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 468270d
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 1290g4, its twist by $33$.
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.