Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-6555733716x+204306139713652\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-6555733716xz^2+204306139713652z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-6555733716x+204306139713652\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(46405816049369/1634584900, 423525331424600374653/66086267507000)$ | $21.483891534805410527568586238$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 50616 \) | = | $2^{3} \cdot 3^{2} \cdot 19 \cdot 37$ |
|
| Discriminant: | $\Delta$ | = | $-111868643777541506421744384$ | = | $-1 \cdot 2^{8} \cdot 3^{14} \cdot 19 \cdot 37^{10} $ |
|
| j-invariant: | $j$ | = | \( -\frac{166962959078001445737309395968}{599433319281236638491} \) | = | $-1 \cdot 2^{10} \cdot 3^{-8} \cdot 7^{6} \cdot 19^{-1} \cdot 37^{-10} \cdot 317^{3} \cdot 35171^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.2159669875692093435580734863$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $3.2045627228618576249156294535$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0868469630613313$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.3323724292041526$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $21.483891534805410527568586238$ |
|
| Real period: | $\Omega$ | ≈ | $0.051924635820749374414921531524$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2\cdot1\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $8.9243459516580101988698028714 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 8.924345952 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.051925 \cdot 21.483892 \cdot 8}{1^2} \\ & \approx 8.924345952\end{aligned}$$
Modular invariants
Modular form 50616.2.a.i
For more coefficients, see the Downloads section to the right.
| Modular degree: | 82513920 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
|
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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{1}^{*}$ | additive | -1 | 3 | 8 | 0 |
| $3$ | $2$ | $I_{8}^{*}$ | additive | -1 | 2 | 14 | 8 |
| $19$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $37$ | $2$ | $I_{10}$ | nonsplit multiplicative | 1 | 1 | 10 | 10 |
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 38.2.0.a.1, level \( 38 = 2 \cdot 19 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 37 & 2 \\ 36 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 21 & 2 \\ 21 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 37 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[38])$ is a degree-$369360$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/38\Z)$.
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 | $2$ | \( 171 = 3^{2} \cdot 19 \) |
| $3$ | additive | $8$ | \( 5624 = 2^{3} \cdot 19 \cdot 37 \) |
| $5$ | good | $2$ | \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 2664 = 2^{3} \cdot 3^{2} \cdot 37 \) |
| $37$ | nonsplit multiplicative | $38$ | \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 50616.i consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 16872.b1, its twist by $-3$.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.76.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.109744.2 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | add | ord | ord | ord | ord | ord | nonsplit | ss | ord | ord | nonsplit | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | 1 | 1 | 1 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | - | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.