Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-118668x-15664912\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-118668xz^2-15664912z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-118668x-15664912\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-188, 0)$ | $0$ | $2$ |
Integral points
\( \left(-188, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 37440 \) | = | $2^{6} \cdot 3^{2} \cdot 5 \cdot 13$ |
|
| Discriminant: | $\Delta$ | = | $941763109847040$ | = | $2^{21} \cdot 3^{12} \cdot 5 \cdot 13^{2} $ |
|
| j-invariant: | $j$ | = | \( \frac{967068262369}{4928040} \) | = | $2^{-3} \cdot 3^{-6} \cdot 5^{-1} \cdot 11^{3} \cdot 13^{-2} \cdot 29^{3} \cdot 31^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7195439833218733579818030508$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.13051706814790054815833225015$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9529644067700619$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.431497114948641$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.25730567534395648268600819531$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot2^{2}\cdot1\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L(E,1)$ | ≈ | $2.0584454027516518614880655625 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 2.058445403 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.257306 \cdot 1.000000 \cdot 32}{2^2} \\ & \approx 2.058445403\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 147456 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{11}^{*}$ | additive | 1 | 6 | 21 | 3 |
| $3$ | $4$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $13$ | $2$ | $I_{2}$ | nonsplit 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 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 10 & 3 \\ 1221 & 1552 \end{array}\right),\left(\begin{array}{rr} 1081 & 12 \\ 246 & 73 \end{array}\right),\left(\begin{array}{rr} 1550 & 1557 \\ 807 & 8 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 511 & 1558 \\ 1296 & 1559 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 1510 & 1551 \end{array}\right),\left(\begin{array}{rr} 1549 & 12 \\ 1548 & 13 \end{array}\right),\left(\begin{array}{rr} 75 & 328 \\ 38 & 317 \end{array}\right)$.
The torsion field $K:=\Q(E[1560])$ is a degree-$9661317120$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1560\Z)$.
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 | $2$ | \( 4160 = 2^{6} \cdot 5 \cdot 13 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 7488 = 2^{6} \cdot 3^{2} \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 37440bg
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 390c2, 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 |
| $2$ | \(\Q(\sqrt{-6}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.243360.4 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-6}, \sqrt{10})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.82255680000.2 | \(\Z/6\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.94758543360000.5 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.3790341734400.38 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.0.15215916481432588745578292812185600000000.2 | \(\Z/18\Z\) | 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 | 13 |
|---|---|---|---|---|
| Reduction type | add | add | nonsplit | nonsplit |
| $\lambda$-invariant(s) | - | - | 0 | 0 |
| $\mu$-invariant(s) | - | - | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ 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 $0$.