Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3+x^2-90x+1255\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3+x^2z-90xz^2+1255z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-116667x+60311574\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{5}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-7, 43\right) \) | $0$ | $5$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-7:43:1]\) | $0$ | $5$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-237, 8640\right) \) | $0$ | $5$ |
Integral points
\( \left(-7, 43\right) \), \( \left(-7, -37\right) \), \( \left(13, 43\right) \), \( \left(13, -57\right) \)
\([-7:43:1]\), \([-7:-37:1]\), \([13:43:1]\), \([13:-57:1]\)
\((-237,\pm 8640)\), \((483,\pm 10800)\)
Invariants
| Conductor: | $N$ | = | \( 2110 \) | = | $2 \cdot 5 \cdot 211$ |
|
| Minimal Discriminant: | $\Delta$ | = | $-675200000$ | = | $-1 \cdot 2^{10} \cdot 5^{5} \cdot 211 $ |
|
| j-invariant: | $j$ | = | \( -\frac{80677568161}{675200000} \) | = | $-1 \cdot 2^{-10} \cdot 5^{-5} \cdot 29^{3} \cdot 149^{3} \cdot 211^{-1}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.37691625425607364081738694451$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.37691625425607364081738694451$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.8969980273752904$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.638684127633288$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $1.3817438784369413754939001538$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 50 $ = $ ( 2 \cdot 5 )\cdot5\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $5$ |
|
| Special value: | $ L(E,1)$ | ≈ | $2.7634877568738827509878003076 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 2.763487757 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.381744 \cdot 1.000000 \cdot 50}{5^2} \\ & \approx 2.763487757\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 960 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $10$ | $I_{10}$ | split multiplicative | -1 | 1 | 10 | 10 |
| $5$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $211$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
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 | $\ell$-adic index |
|---|---|---|---|
| $5$ | 5B.1.1 | 5.24.0.1 | $24$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2110 = 2 \cdot 5 \cdot 211 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 6 & 13 \\ 2055 & 1991 \end{array}\right),\left(\begin{array}{rr} 2101 & 10 \\ 2100 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1901 & 10 \\ 1065 & 51 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 5 & 1317 \end{array}\right)$.
The torsion field $K:=\Q(E[2110])$ is a degree-$118360872000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2110\Z)$.
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 | $4$ | \( 1055 = 5 \cdot 211 \) |
| $5$ | split multiplicative | $6$ | \( 211 \) |
| $211$ | split multiplicative | $212$ | \( 10 = 2 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 2110.f
consists of 2 curves linked by isogenies of
degree 5.
Twists
This elliptic curve is its own minimal quadratic twist.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{5}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.1055.1 | \(\Z/10\Z\) | not in database |
| $6$ | 6.0.1174241375.1 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $8$ | deg 8 | \(\Z/15\Z\) | not in database |
| $12$ | deg 12 | \(\Z/20\Z\) | not in database |
| $20$ | 20.0.471052539862207661466330102369093486358642578125.2 | \(\Z/5\Z \oplus \Z/5\Z\) | 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 | 211 |
|---|---|---|---|---|
| Reduction type | split | ord | split | split |
| $\lambda$-invariant(s) | 8 | 0 | 1 | 1 |
| $\mu$-invariant(s) | 0 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.