Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+y=x^3-x^2-1160x-35745\)
|
(homogenize, simplify) |
|
\(y^2z+yz^2=x^3-x^2z-1160xz^2-35745z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-1503792x-1685750832\)
|
(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 38291 \) | = | $11 \cdot 59^{2}$ |
|
| Discriminant: | $\Delta$ | = | $-463985870051$ | = | $-1 \cdot 11 \cdot 59^{6} $ |
|
| j-invariant: | $j$ | = | \( -\frac{4096}{11} \) | = | $-1 \cdot 2^{12} \cdot 11^{-1}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.92603992461740647331863125879$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.1127287973354532519893939281$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.8254556483942886$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.270452316254111$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.37984349335990477395851542547$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L(E,1)$ | ≈ | $0.37984349335990477395851542547 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 0.379843493 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.379843 \cdot 1.000000 \cdot 1}{1^2} \\ & \approx 0.379843493\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 41470 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $11$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $59$ | $1$ | $I_0^{*}$ | 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 |
|---|---|---|
| $5$ | 5B.4.1 | 25.60.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 32450 = 2 \cdot 5^{2} \cdot 11 \cdot 59 \), index $1200$, genus $37$, and generators
$\left(\begin{array}{rr} 17049 & 0 \\ 0 & 32449 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 50 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 50 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 16816 & 12095 \\ 3127 & 9087 \end{array}\right),\left(\begin{array}{rr} 12391 & 24780 \\ 21535 & 8261 \end{array}\right),\left(\begin{array}{rr} 38 & 41 \\ 29891 & 29689 \end{array}\right),\left(\begin{array}{rr} 32401 & 50 \\ 32400 & 51 \end{array}\right)$.
The torsion field $K:=\Q(E[32450])$ is a degree-$235789488000000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/32450\Z)$.
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 |
|---|---|---|---|
| $11$ | nonsplit multiplicative | $12$ | \( 3481 = 59^{2} \) |
| $59$ | additive | $1742$ | \( 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5 and 25.
Its isogeny class 38291c
consists of 3 curves linked by isogenies of
degrees dividing 25.
Twists
The minimal quadratic twist of this elliptic curve is 11a3, its twist by $-59$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-59}) \) | \(\Z/5\Z\) | not in database |
| $3$ | 3.1.44.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.21296.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.397613744.3 | \(\Z/10\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $10$ | 10.0.153250372733349419.1 | \(\Z/25\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $16$ | deg 16 | \(\Z/15\Z\) | not in database |
| $20$ | 20.4.716725974820267549843417552112962677001953125.1 | \(\Z/5\Z\) | 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 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | ss | ord | ord | ord | nonsplit | ord | ord | ss | ord | ss | ord | ord | ord | ord | ord | add |
| $\lambda$-invariant(s) | 0,1 | 0 | 2 | 0 | 0 | 0 | 0 | 0,0 | 2 | 0,0 | 0 | 0 | 0 | 0 | 0 | - |
| $\mu$-invariant(s) | 0,0 | 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
All $p$-adic regulators are identically $1$ since the rank is $0$.