Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3-x^2-2510254697x+60874200081321\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3-x^2z-2510254697xz^2+60874200081321z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-40164075147x+3895908641129414\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-59449, 29724)$ | $0$ | $2$ |
Integral points
\( \left(-59449, 29724\right) \)
Invariants
| Conductor: | $N$ | = | \( 48510 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 11$ |
|
| Discriminant: | $\Delta$ | = | $-588459489751677219779484057600$ | = | $-1 \cdot 2^{38} \cdot 3^{13} \cdot 5^{2} \cdot 7^{9} \cdot 11^{3} $ |
|
| j-invariant: | $j$ | = | \( -\frac{59465789423385795028207}{20003531867239219200} \) | = | $-1 \cdot 2^{-38} \cdot 3^{-7} \cdot 5^{-2} \cdot 11^{-3} \cdot 163^{3} \cdot 239461^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.4242026983278591326471203467$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.4154639422023193081204831707$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.043332889994024$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.136809480606234$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.027391241223057661156732169981$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 912 $ = $ ( 2 \cdot 19 )\cdot2\cdot2\cdot2\cdot3 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L(E,1)$ | ≈ | $6.2452029988571467437349347557 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 6.245202999 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.027391 \cdot 1.000000 \cdot 912}{2^2} \\ & \approx 6.245202999\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 85800960 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $38$ | $I_{38}$ | split multiplicative | -1 | 1 | 38 | 38 |
| $3$ | $2$ | $I_{7}^{*}$ | additive | -1 | 2 | 13 | 7 |
| $5$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $7$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 0 |
| $11$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
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 \( 9240 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 2644 & 1 \\ 3959 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4621 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 3082 & 1 \\ 3079 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 8089 & 1156 \\ 3464 & 5775 \end{array}\right),\left(\begin{array}{rr} 5042 & 1 \\ 6719 & 0 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 3697 & 4 \\ 7394 & 9 \end{array}\right),\left(\begin{array}{rr} 9237 & 4 \\ 9236 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[9240])$ is a degree-$78479622144000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9240\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 693 = 3^{2} \cdot 7 \cdot 11 \) |
| $3$ | additive | $8$ | \( 490 = 2 \cdot 5 \cdot 7^{2} \) |
| $5$ | split multiplicative | $6$ | \( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \) |
| $7$ | additive | $20$ | \( 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 \) |
| $11$ | split multiplicative | $12$ | \( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \) |
| $19$ | good | $2$ | \( 24255 = 3^{2} \cdot 5 \cdot 7^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 48510.em
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 16170.x2, its twist by $21$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-231}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.2.18110400.5 | \(\Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.208411674030000.10 | \(\Z/6\Z\) | 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 |
|---|---|---|---|---|---|---|---|---|
| Reduction type | split | add | split | add | split | ord | ord | ord |
| $\lambda$-invariant(s) | 6 | - | 3 | - | 1 | 0 | 0 | 2 |
| $\mu$-invariant(s) | 0 | - | 0 | - | 0 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 23$ 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 $0$.