Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-x^2-9008x-313488\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-x^2z-9008xz^2-313488z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-729675x-230721750\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(337, 5900)$ | $4.3594310528224842554478189474$ | $\infty$ |
| $(-63, 0)$ | $0$ | $2$ |
Integral points
\( \left(-63, 0\right) \), \((337,\pm 5900)\)
Invariants
| Conductor: | $N$ | = | \( 13200 \) | = | $2^{4} \cdot 3 \cdot 5^{2} \cdot 11$ |
|
| Discriminant: | $\Delta$ | = | $3513840000000$ | = | $2^{10} \cdot 3 \cdot 5^{7} \cdot 11^{4} $ |
|
| j-invariant: | $j$ | = | \( \frac{5052857764}{219615} \) | = | $2^{2} \cdot 3^{-1} \cdot 5^{-1} \cdot 11^{-4} \cdot 23^{3} \cdot 47^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.1708472038024250563375106965$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.21149440288124622214389573799$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9023760487772444$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.10322866137629$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.3594310528224842554478189474$ |
|
| Real period: | $\Omega$ | ≈ | $0.49136420496758744311893618164$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot1\cdot2\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $4.2841367467622653506688995556 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 4.284136747 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.491364 \cdot 4.359431 \cdot 8}{2^2} \\ & \approx 4.284136747\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 24576 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}^{*}$ | additive | 1 | 4 | 10 | 0 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $2$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $11$ | $2$ | $I_{4}$ | 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 \( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1052 & 1319 \\ 769 & 1314 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 1201 & 8 \\ 844 & 33 \end{array}\right),\left(\begin{array}{rr} 1152 & 487 \\ 1127 & 1080 \end{array}\right),\left(\begin{array}{rr} 1313 & 8 \\ 1312 & 9 \end{array}\right),\left(\begin{array}{rr} 499 & 498 \\ 178 & 835 \end{array}\right),\left(\begin{array}{rr} 884 & 1 \\ 463 & 6 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 1314 & 1315 \end{array}\right)$.
The torsion field $K:=\Q(E[1320])$ is a degree-$9732096000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1320\Z)$.
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 | $4$ | \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \) |
| $5$ | additive | $18$ | \( 528 = 2^{4} \cdot 3 \cdot 11 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 13200.s
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1320.g1, its twist by $-20$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{15}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-5}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{-5})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.2916000000.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.5397258240000.23 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.683090496000000.15 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | 16.0.8503056000000000000.1 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | nonsplit | add | ss | nonsplit | ord | ord | ss | ss | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 1 | - | 1,1 | 1 | 1 | 1 | 1,1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1,3 |
| $\mu$-invariant(s) | - | 0 | - | 0,0 | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.