Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3+11833437x-3407756638\)
|
(homogenize, simplify) |
|
\(y^2z=x^3+11833437xz^2-3407756638z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+11833437x-3407756638\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1405559/169, 1798725050/2197)$ | $12.060969133696727558985331951$ | $\infty$ |
| $(286, 0)$ | $0$ | $2$ |
Integral points
\( \left(286, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 87120 \) | = | $2^{4} \cdot 3^{2} \cdot 5 \cdot 11^{2}$ |
|
| Discriminant: | $\Delta$ | = | $-111067222732265226240000$ | = | $-1 \cdot 2^{21} \cdot 3^{3} \cdot 5^{4} \cdot 11^{12} $ |
|
| j-invariant: | $j$ | = | \( \frac{935355271080573}{566899520000} \) | = | $2^{-9} \cdot 3^{3} \cdot 5^{-4} \cdot 7^{3} \cdot 11^{-6} \cdot 4657^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.1112646680452969545775368857$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.94451677891913895028052166603$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0709081254433794$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.31628133770948$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $12.060969133696727558985331951$ |
|
| Real period: | $\Omega$ | ≈ | $0.061236676253613634236366411470$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2\cdot2\cdot2^{2} $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $5.9085892971601072175945806766 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 5.908589297 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.061237 \cdot 12.060969 \cdot 32}{2^2} \\ & \approx 5.908589297\end{aligned}$$
Modular invariants
Modular form 87120.2.a.g
For more coefficients, see the Downloads section to the right.
| Modular degree: | 9953280 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{13}^{*}$ | additive | -1 | 4 | 21 | 9 |
| $3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $5$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $11$ | $4$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
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 \( 264 = 2^{3} \cdot 3 \cdot 11 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 10 & 3 \\ 105 & 256 \end{array}\right),\left(\begin{array}{rr} 253 & 12 \\ 252 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 119 & 252 \\ 186 & 191 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 214 & 255 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 56 & 11 \\ 141 & 232 \end{array}\right),\left(\begin{array}{rr} 45 & 8 \\ 82 & 19 \end{array}\right)$.
The torsion field $K:=\Q(E[264])$ is a degree-$10137600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/264\Z)$.
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 | $4$ | \( 363 = 3 \cdot 11^{2} \) |
| $3$ | additive | $6$ | \( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 17424 = 2^{4} \cdot 3^{2} \cdot 11^{2} \) |
| $11$ | additive | $72$ | \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 87120dd
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 990b4, its twist by $-132$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-6}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{11}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.2.26136.2 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-6}, \sqrt{11})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.116435880000.10 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.179068074983424.3 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.43717791744.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.43717791744.10 | \(\Z/12\Z\) | 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 |
| $18$ | 18.6.88069975515267944949140884633497600000000.2 | \(\Z/18\Z\) | 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 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | add | nonsplit | ord | add | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | 1 | 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 |
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.