Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3+x^2-53x-237\)
|
(homogenize, simplify) |
|
\(y^2z=x^3+x^2z-53xz^2-237z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-4320x-159840\)
|
(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 4800 \) | = | $2^{6} \cdot 3 \cdot 5^{2}$ |
|
| Minimal Discriminant: | $\Delta$ | = | $-11059200$ | = | $-1 \cdot 2^{14} \cdot 3^{3} \cdot 5^{2} $ |
|
| j-invariant: | $j$ | = | \( -\frac{40960}{27} \) | = | $-1 \cdot 2^{13} \cdot 3^{-3} \cdot 5$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.052484306877396647329693894065$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.0244270558482229427572034698$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.029908311854535$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.8672240392510737$ | |||
| 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: | $\Omega$ | ≈ | $0.85798714531928150501018134576$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 3 $ = $ 1\cdot3\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L(E,1)$ | ≈ | $2.5739614359578445150305440373 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 2.573961436 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.857987 \cdot 1.000000 \cdot 3}{1^2} \\ & \approx 2.573961436\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1152 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II^{*}$ | additive | 1 | 6 | 14 | 0 |
| $3$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $5$ | $1$ | $II$ | additive | 1 | 2 | 2 | 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 | $\ell$-adic index |
|---|---|---|---|
| $3$ | 3B | 3.4.0.1 | $4$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 120 = 2^{3} \cdot 3 \cdot 5 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 31 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 21 & 16 \\ 25 & 21 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 59 & 0 \\ 0 & 119 \end{array}\right),\left(\begin{array}{rr} 115 & 6 \\ 114 & 7 \end{array}\right),\left(\begin{array}{rr} 25 & 6 \\ 24 & 7 \end{array}\right),\left(\begin{array}{rr} 115 & 114 \\ 102 & 91 \end{array}\right),\left(\begin{array}{rr} 2 & 5 \\ 1 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[120])$ is a degree-$737280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/120\Z)$.
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 | $2$ | \( 75 = 3 \cdot 5^{2} \) |
| $3$ | split multiplicative | $4$ | \( 1600 = 2^{6} \cdot 5^{2} \) |
| $5$ | additive | $10$ | \( 192 = 2^{6} \cdot 3 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 4800s
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 300a1, its twist by $8$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{10}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.300.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.270000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.172800000.1 | \(\Z/6\Z\) | not in database |
| $6$ | 6.2.57600000.1 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.6.53973124931819667456000000000000000.3 | \(\Z/9\Z\) | 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 | split | add | ord | ord | ord | ord | ord | ord | ord | ord | ord | ss | ord | ord |
| $\lambda$-invariant(s) | - | 3 | - | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 |
| $\mu$-invariant(s) | - | 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 $0$.