Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-40156428x-80613670448\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-40156428xz^2-80613670448z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-40156428x-80613670448\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(10757, 855657)$ | $6.5871358928019254713925641079$ | $\infty$ |
| $(939932, 911244168)$ | $7.0218144066126872433398553160$ | $\infty$ |
| $(-4852, 0)$ | $0$ | $2$ |
Integral points
\( \left(-4852, 0\right) \), \((10757,\pm 855657)\), \((939932,\pm 911244168)\)
Invariants
| Conductor: | $N$ | = | \( 123840 \) | = | $2^{6} \cdot 3^{2} \cdot 5 \cdot 43$ |
|
| Discriminant: | $\Delta$ | = | $1336863265160624640000000$ | = | $2^{15} \cdot 3^{24} \cdot 5^{7} \cdot 43^{2} $ |
|
| j-invariant: | $j$ | = | \( \frac{299786086083570891272}{55964100325078125} \) | = | $2^{3} \cdot 3^{-18} \cdot 5^{-7} \cdot 13^{6} \cdot 43^{-2} \cdot 19801^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.3473921945922639763409588119$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.9316520745582774938717960416$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.074091800502521$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.469418710884038$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
|
| Mordell-Weil rank: | $r$ | = | $ 2$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $45.893461967926177577765753901$ |
|
| Real period: | $\Omega$ | ≈ | $0.060747767129900881091584118431$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2^{2}\cdot1\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $11.151701361650168218685819460 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 11.151701362 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.060748 \cdot 45.893462 \cdot 16}{2^2} \\ & \approx 11.151701362\end{aligned}$$
Modular invariants
Modular form 123840.2.a.w
For more coefficients, see the Downloads section to the right.
| Modular degree: | 14450688 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{5}^{*}$ | additive | -1 | 6 | 15 | 0 |
| $3$ | $4$ | $I_{18}^{*}$ | additive | -1 | 2 | 24 | 18 |
| $5$ | $1$ | $I_{7}$ | nonsplit multiplicative | 1 | 1 | 7 | 7 |
| $43$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 5160 = 2^{3} \cdot 3 \cdot 5 \cdot 43 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1034 & 1 \\ 2063 & 0 \end{array}\right),\left(\begin{array}{rr} 4561 & 4 \\ 3962 & 9 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 2579 & 0 \end{array}\right),\left(\begin{array}{rr} 1937 & 3226 \\ 3224 & 1935 \end{array}\right),\left(\begin{array}{rr} 5157 & 4 \\ 5156 & 5 \end{array}\right),\left(\begin{array}{rr} 1721 & 4 \\ 3442 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[5160])$ is a degree-$9842652610560$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5160\Z)$.
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 | $4$ | \( 45 = 3^{2} \cdot 5 \) |
| $3$ | additive | $6$ | \( 13760 = 2^{6} \cdot 5 \cdot 43 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 24768 = 2^{6} \cdot 3^{2} \cdot 43 \) |
| $7$ | good | $2$ | \( 24768 = 2^{6} \cdot 3^{2} \cdot 43 \) |
| $43$ | split multiplicative | $44$ | \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 123840.w
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 20640.b1, its twist by $-24$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{10}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.2662560.1 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.11342761205760000.60 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | split | ord |
| $\lambda$-invariant(s) | - | - | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 2 |
| $\mu$-invariant(s) | - | - | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.