Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-x^2-93175008x-346008031488\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-x^2z-93175008xz^2-346008031488z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-7547175675x-252262496481750\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-\frac{2921880646977211}{515980749124}, \frac{309533792215504386629}{370638259749253432}\right) \) | $34.433838297173465644601710760$ | $\infty$ |
| \( \left(-5663, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-2098839462575376251098:309533792215504386629:370638259749253432]\) | $34.433838297173465644601710760$ | $\infty$ |
| \([-5663:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-\frac{26298473765042271}{515980749124}, \frac{8357412389818618438983}{370638259749253432}\right) \) | $34.433838297173465644601710760$ | $\infty$ |
| \( \left(-50970, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-5663, 0\right) \)
\([-5663:0:1]\)
\( \left(-5663, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 418800 \) | = | $2^{4} \cdot 3 \cdot 5^{2} \cdot 349$ |
|
| Minimal Discriminant: | $\Delta$ | = | $41017145060044800000000$ | = | $2^{18} \cdot 3^{3} \cdot 5^{8} \cdot 349^{4} $ |
|
| j-invariant: | $j$ | = | \( \frac{1397790417785316543001}{640892891563200} \) | = | $2^{-6} \cdot 3^{-3} \cdot 5^{-2} \cdot 13^{3} \cdot 349^{-4} \cdot 860077^{3}$ |
|
| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
|
||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $3.2959937646325708061859152704$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.7981276278555753094683034823$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9771020392492882$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.149695423155166$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $34.433838297173465644601710760$ |
|
| Real period: | $\Omega$ | ≈ | $0.048594447153479550085333195385$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot1\cdot2\cdot2^{2} $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $6.6931733416938249425368715999 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 6.693173342 \approx L'(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.048594 \cdot 34.433838 \cdot 16}{2^2} \\ & \approx 6.693173342\end{aligned}$$
Modular invariants
Modular form 418800.2.a.c
For more coefficients, see the Downloads section to the right.
| Modular degree: | 71663616 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 (conditional*) |
|
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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{10}^{*}$ | additive | -1 | 4 | 18 | 6 |
| $3$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $5$ | $2$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
| $349$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 4.6.0.1 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 41880 = 2^{3} \cdot 3 \cdot 5 \cdot 349 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 33503 & 0 \\ 0 & 41879 \end{array}\right),\left(\begin{array}{rr} 30716 & 33505 \\ 2815 & 6 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 41874 & 41875 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 3841 & 16760 \\ 23740 & 25161 \end{array}\right),\left(\begin{array}{rr} 21984 & 9415 \\ 30335 & 38664 \end{array}\right),\left(\begin{array}{rr} 41873 & 8 \\ 41872 & 9 \end{array}\right),\left(\begin{array}{rr} 1051 & 1050 \\ 11530 & 24091 \end{array}\right)$.
The torsion field $K:=\Q(E[41880])$ is a degree-$10906475102208000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/41880\Z)$.
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 | $2$ | \( 75 = 3 \cdot 5^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 139600 = 2^{4} \cdot 5^{2} \cdot 349 \) |
| $5$ | additive | $18$ | \( 16752 = 2^{4} \cdot 3 \cdot 349 \) |
| $349$ | split multiplicative | $350$ | \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 418800.c
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 10470.a1, 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.