Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2+85159x+434928420\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z+85159xz^2+434928420z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+6897852x+317042124597\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 17340 \) | = | $2^{2} \cdot 3 \cdot 5 \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $-81747157511718750000$ | = | $-1 \cdot 2^{4} \cdot 3 \cdot 5^{12} \cdot 17^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{611926016}{732421875} \) | = | $2^{14} \cdot 3^{-1} \cdot 5^{-12} \cdot 13^{3} \cdot 17$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4998400412766705627928630102$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.37998208505254473948742922446$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1479077211288085$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.4610847525068715$ | |||
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: | $\Omega$ | ≈ | $0.15049583461655594665173604700$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot1\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.7089250230980070397312488460 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $9$ = $3^2$ (exact) |
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BSD formula
$$\begin{aligned} 2.708925023 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{9 \cdot 0.150496 \cdot 1.000000 \cdot 2}{1^2} \\ & \approx 2.708925023\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 396576 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $3$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $5$ | $2$ | $I_{12}$ | nonsplit multiplicative | 1 | 1 | 12 | 12 |
| $17$ | $1$ | $IV^{*}$ | additive | -1 | 2 | 8 | 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 |
|---|---|---|
| $3$ | 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
$\left(\begin{array}{rr} 2 & 3 \\ 1 & 4 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 3 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 2 & 5 \end{array}\right)$.
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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 867 = 3 \cdot 17^{2} \) |
| $3$ | split multiplicative | $4$ | \( 1156 = 2^{2} \cdot 17^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 3468 = 2^{2} \cdot 3 \cdot 17^{2} \) |
| $17$ | additive | $114$ | \( 60 = 2^{2} \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 17340.k
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 17340.h2, its twist by $17$.
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 |
| $3$ | 3.1.867.1 | \(\Z/2\Z\) | not in database |
| $3$ | 3.1.280908.3 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.2255067.2 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.236727913392.1 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $9$ | 9.1.66498764695119936.17 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $18$ | 18.0.359870256020257945533034946667000000000000.2 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.13266257117930788904129800273932288.2 | \(\Z/6\Z \oplus \Z/6\Z\) | 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 | 5 | 17 |
|---|---|---|---|---|
| Reduction type | add | split | nonsplit | add |
| $\lambda$-invariant(s) | - | 5 | 0 | - |
| $\mu$-invariant(s) | - | 1 | 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 $0$.