Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2+25x-161\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z+25xz^2-161z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+405x-9882\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(7, 14\right) \) | $0$ | $3$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([7:14:1]\) | $0$ | $3$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(27, 144\right) \) | $0$ | $3$ |
Integral points
\( \left(7, 14\right) \), \( \left(7, -22\right) \)
\([7:14:1]\), \([7:-22:1]\)
\((27,\pm 144)\)
Invariants
| Conductor: | $N$ | = | \( 486 \) | = | $2 \cdot 3^{5}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-11337408$ | = | $-1 \cdot 2^{6} \cdot 3^{11} $ |
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| j-invariant: | $j$ | = | \( \frac{10125}{64} \) | = | $2^{-6} \cdot 3^{4} \cdot 5^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $0.035446204425096299910847722428$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.97161506018733758386812707808$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1072533652194176$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.815312609812786$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $1.1331617992062495436141801263$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 18 $ = $ ( 2 \cdot 3 )\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $3$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.2663235984124990872283602526 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 2.266323598 \approx L(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.133162 \cdot 1.000000 \cdot 18}{3^2} \\ & \approx 2.266323598\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 108 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $3$ | $3$ | $IV^{*}$ | additive | 1 | 5 | 11 | 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.1.1 | 3.8.0.1 | $8$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 6.48.0-6.c.1.2, level \( 6 = 2 \cdot 3 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 2 & 5 \\ 1 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 1 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[6])$ is a degree-$6$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6\Z)$.
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 | $4$ | \( 243 = 3^{5} \) |
| $3$ | additive | $8$ | \( 1 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 486d
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 486a1, its twist by $-3$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | \(\Q(\sqrt[3]{3})\) | \(\Z/6\Z\) | not in database |
| $6$ | 6.0.177147.2 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $9$ | 9.3.669462604992.1 | \(\Z/9\Z\) | not in database |
| $12$ | 12.2.6169767367606272.2 | \(\Z/12\Z\) | not in database |
We only show fields where the torsion growth is primitive.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | split | add | ss | ord | ord | ord | ord | ord | ord | ord | ord | ord | ss | ord | ord |
| $\lambda$-invariant(s) | 1 | - | 0,0 | 0 | 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 | 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 $0$.