Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-402204x-14012768\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-402204xz^2-14012768z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-402204x-14012768\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1496, 52272)$ | $2.4750550277793464273306257272$ | $\infty$ |
| $(-616, 0)$ | $0$ | $2$ |
Integral points
\( \left(-616, 0\right) \), \((1496,\pm 52272)\)
Invariants
| Conductor: | $N$ | = | \( 453024 \) | = | $2^{5} \cdot 3^{2} \cdot 11^{2} \cdot 13$ |
|
| Discriminant: | $\Delta$ | = | $4079254117951254528$ | = | $2^{12} \cdot 3^{9} \cdot 11^{6} \cdot 13^{4} $ |
|
| j-invariant: | $j$ | = | \( \frac{1360251712}{771147} \) | = | $2^{6} \cdot 3^{-3} \cdot 13^{-4} \cdot 277^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2609701230826214881241773563$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.18043083821056393902164917260$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.069422794221752$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.8643200977183727$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.4750550277793464273306257272$ |
|
| Real period: | $\Omega$ | ≈ | $0.20458667294070998193390301704$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2\cdot2^{2}\cdot2^{2}\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $8.1018123756568481019261979011 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 8.101812376 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.204587 \cdot 2.475055 \cdot 64}{2^2} \\ & \approx 8.101812376\end{aligned}$$
Modular invariants
Modular form 453024.2.a.dc
For more coefficients, see the Downloads section to the right.
| Modular degree: | 7864320 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{3}^{*}$ | additive | 1 | 5 | 12 | 0 |
| $3$ | $4$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
| $11$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $13$ | $2$ | $I_{4}$ | nonsplit 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 |
|---|---|---|
| $2$ | 2B | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3432 = 2^{3} \cdot 3 \cdot 11 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 7 & 6 \\ 3426 & 3427 \end{array}\right),\left(\begin{array}{rr} 3425 & 8 \\ 3424 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 3323 & 198 \\ 2618 & 1211 \end{array}\right),\left(\begin{array}{rr} 1440 & 1903 \\ 671 & 1308 \end{array}\right),\left(\begin{array}{rr} 2641 & 2816 \\ 3388 & 969 \end{array}\right),\left(\begin{array}{rr} 1660 & 935 \\ 2057 & 1242 \end{array}\right),\left(\begin{array}{rr} 2495 & 0 \\ 0 & 3431 \end{array}\right)$.
The torsion field $K:=\Q(E[3432])$ is a degree-$531372441600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3432\Z)$.
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 | $2$ | \( 1089 = 3^{2} \cdot 11^{2} \) |
| $3$ | additive | $6$ | \( 50336 = 2^{5} \cdot 11^{2} \cdot 13 \) |
| $11$ | additive | $62$ | \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 34848 = 2^{5} \cdot 3^{2} \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 453024dc
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1248h3, its twist by $-132$.
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.