Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-x^2-5633x+155137\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-x^2z-5633xz^2+155137z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-456300x+111726000\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(33, 64)$ | $1.4291068523418990735818669007$ | $\infty$ |
Integral points
\((33,\pm 64)\), \((296,\pm 4933)\)
Invariants
| Conductor: | $N$ | = | \( 126400 \) | = | $2^{6} \cdot 5^{2} \cdot 79$ |
|
| Discriminant: | $\Delta$ | = | $1294336000000$ | = | $2^{20} \cdot 5^{6} \cdot 79 $ |
|
| j-invariant: | $j$ | = | \( \frac{4826809}{316} \) | = | $2^{-2} \cdot 13^{6} \cdot 79^{-1}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.0738746064561375991616936495$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.77056512060083055226453419930$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9406342793009744$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.1942038336297407$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.4291068523418990735818669007$ |
|
| Real period: | $\Omega$ | ≈ | $0.84394623132962809027575666476$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2^{2}\cdot1\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $4.8243573688051720362770647944 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 4.824357369 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.843946 \cdot 1.429107 \cdot 4}{1^2} \\ & \approx 4.824357369\end{aligned}$$
Modular invariants
Modular form 126400.2.a.bf
For more coefficients, see the Downloads section to the right.
| Modular degree: | 215040 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{10}^{*}$ | additive | 1 | 6 | 20 | 2 |
| $5$ | $1$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $79$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
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 \( 316 = 2^{2} \cdot 79 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 1 \\ 315 & 0 \end{array}\right),\left(\begin{array}{rr} 315 & 2 \\ 314 & 3 \end{array}\right),\left(\begin{array}{rr} 161 & 2 \\ 161 & 3 \end{array}\right),\left(\begin{array}{rr} 159 & 2 \\ 159 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[316])$ is a degree-$1845642240$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/316\Z)$.
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 | $2$ | \( 1975 = 5^{2} \cdot 79 \) |
| $5$ | additive | $14$ | \( 5056 = 2^{6} \cdot 79 \) |
| $79$ | nonsplit multiplicative | $80$ | \( 1600 = 2^{6} \cdot 5^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 126400.bf consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 158.a1, its twist by $40$.
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 |
|---|---|---|---|
| $3$ | 3.3.316.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.6.31554496.1 | \(\Z/2\Z \oplus \Z/2\Z\) | 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 | 79 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | ord | add | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | nonsplit |
| $\lambda$-invariant(s) | - | 1 | - | 3 | 3 | 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.