Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3-76900x-8214415\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3-76900xz^2-8214415z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-99662427x-382952758986\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-641/4, 641/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 14415 \) | = | $3 \cdot 5 \cdot 31^{2}$ |
|
| Discriminant: | $\Delta$ | = | $13312555215$ | = | $3 \cdot 5 \cdot 31^{6} $ |
|
| j-invariant: | $j$ | = | \( \frac{56667352321}{15} \) | = | $3^{-1} \cdot 5^{-1} \cdot 23^{3} \cdot 167^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.3147160301597293355209856651$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.40227757208284378744359649717$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.030187200332639$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.737288087609158$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.28669356641386928815125078469$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot1\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L(E,1)$ | ≈ | $2.2935485313109543052100062775 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $16$ = $4^2$ (exact) |
|
BSD formula
$$\begin{aligned} 2.293548531 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{16 \cdot 0.286694 \cdot 1.000000 \cdot 2}{2^2} \\ & \approx 2.293548531\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 30720 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $31$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
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 | 32.48.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 14880 = 2^{5} \cdot 3 \cdot 5 \cdot 31 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 11223 & 1922 \\ 14074 & 12463 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 28 \\ 68 & 381 \end{array}\right),\left(\begin{array}{rr} 8402 & 3379 \\ 14415 & 9518 \end{array}\right),\left(\begin{array}{rr} 1919 & 0 \\ 0 & 14879 \end{array}\right),\left(\begin{array}{rr} 23 & 18 \\ 12318 & 12875 \end{array}\right),\left(\begin{array}{rr} 12990 & 31 \\ 13609 & 12804 \end{array}\right),\left(\begin{array}{rr} 14849 & 32 \\ 14848 & 33 \end{array}\right),\left(\begin{array}{rr} 11254 & 2883 \\ 8525 & 3628 \end{array}\right)$.
The torsion field $K:=\Q(E[14880])$ is a degree-$10531897344000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/14880\Z)$.
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 |
|---|---|---|---|
| $3$ | split multiplicative | $4$ | \( 4805 = 5 \cdot 31^{2} \) |
| $5$ | split multiplicative | $6$ | \( 2883 = 3 \cdot 31^{2} \) |
| $31$ | additive | $482$ | \( 15 = 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4, 8 and 16.
Its isogeny class 14415g
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 15a7, its twist by $-31$.
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 |
| $2$ | \(\Q(\sqrt{-465}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-31}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{15}, \sqrt{-31})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{-31})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{5}, \sqrt{-31})\) | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.11968832160000.5 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.3694084000000.1 | \(\Z/16\Z\) | not in database |
| $8$ | 8.0.1168831265625.1 | \(\Z/16\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/32\Z\) | 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 | 31 |
|---|---|---|---|---|
| Reduction type | ord | split | split | add |
| $\lambda$-invariant(s) | 16 | 1 | 1 | - |
| $\mu$-invariant(s) | 1 | 0 | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.