Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-164763897x-813991313683\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-164763897xz^2-813991313683z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2636222355x-52098080298066\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 149454 \) | = | $2 \cdot 3^{2} \cdot 19^{2} \cdot 23$ |
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| Discriminant: | $\Delta$ | = | $-724328744291309924352$ | = | $-1 \cdot 2^{12} \cdot 3^{9} \cdot 19^{8} \cdot 23^{2} $ |
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| j-invariant: | $j$ | = | \( -\frac{1479754875499875}{2166784} \) | = | $-1 \cdot 2^{-12} \cdot 3^{6} \cdot 5^{3} \cdot 13^{3} \cdot 19 \cdot 23^{-2} \cdot 73^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.2725264448381506683373770377$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.48560790889277475978492482208$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9838548563022809$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.7385771943778$ | |||
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: | $\Omega$ | ≈ | $0.021069489200858319035637202174$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 24 $ = $ 2\cdot2\cdot3\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.50566774082059965685529285216 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.505667741 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.021069 \cdot 1.000000 \cdot 24}{1^2} \\ & \approx 0.505667741\end{aligned}$$
Modular invariants
Modular form 149454.2.a.t
For more coefficients, see the Downloads section to the right.
| Modular degree: | 19699200 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{12}$ | nonsplit multiplicative | 1 | 1 | 12 | 12 |
| $3$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 0 |
| $19$ | $3$ | $IV^{*}$ | additive | 1 | 2 | 8 | 0 |
| $23$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 |
|---|---|---|
| $3$ | 3B.1.2 | 3.8.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 6.16.0-6.b.1.1, level \( 6 = 2 \cdot 3 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 0 & 5 \\ 5 & 0 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 3 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 2 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[6])$ is a degree-$18$ 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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 1083 = 3 \cdot 19^{2} \) |
| $3$ | additive | $2$ | \( 8303 = 19^{2} \cdot 23 \) |
| $19$ | additive | $146$ | \( 414 = 2 \cdot 3^{2} \cdot 23 \) |
| $23$ | split multiplicative | $24$ | \( 6498 = 2 \cdot 3^{2} \cdot 19^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 149454cr
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 149454cq1, its twist by $57$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.1083.1 | \(\Z/2\Z\) | not in database |
| $3$ | 3.1.5156163.2 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.3518667.2 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.79758050647707.1 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $9$ | 9.1.411245509701832868241.1 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $18$ | 18.0.438515293085316428546163421482683068838817792.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.507368607749760935273908002844166599302243.1 | \(\Z/6\Z \oplus \Z/6\Z\) | 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 | 19 | 23 |
|---|---|---|---|---|
| Reduction type | nonsplit | add | add | split |
| $\lambda$-invariant(s) | 4 | - | - | 1 |
| $\mu$-invariant(s) | 0 | - | - | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ 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 $0$.