Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3+x^2+40942x-1629706\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3+x^2z+40942xz^2-1629706z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+53060400x-76672278000\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 21675 \) | = | $3 \cdot 5^{2} \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $-5558806710796875$ | = | $-1 \cdot 3 \cdot 5^{6} \cdot 17^{9} $ |
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| j-invariant: | $j$ | = | \( \frac{4096}{3} \) | = | $2^{12} \cdot 3^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7094812507516318764559229312$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.2201477135075803710316076988$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9501567537976221$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.3543354086566035$ | |||
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: | $\Omega$ | ≈ | $0.24016026239450870310251255720$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.48032052478901740620502511439 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.480320525 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.240160 \cdot 1.000000 \cdot 2}{1^2} \\ & \approx 0.480320525\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 176256 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $5$ | $1$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $17$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 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 |
|---|---|---|
| $5$ | 5B | 5.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 501 & 10 \\ 500 & 11 \end{array}\right),\left(\begin{array}{rr} 509 & 500 \\ 0 & 203 \end{array}\right),\left(\begin{array}{rr} 6 & 5 \\ 335 & 506 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 455 & 391 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 112 & 505 \\ 235 & 488 \end{array}\right)$.
The torsion field $K:=\Q(E[510])$ is a degree-$225607680$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/510\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 1275 = 3 \cdot 5^{2} \cdot 17 \) |
| $3$ | split multiplicative | $4$ | \( 7225 = 5^{2} \cdot 17^{2} \) |
| $5$ | additive | $14$ | \( 867 = 3 \cdot 17^{2} \) |
| $17$ | additive | $98$ | \( 75 = 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 21675r
consists of 2 curves linked by isogenies of
degree 5.
Twists
The minimal quadratic twist of this elliptic curve is 867e1, its twist by $85$.
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 |
|---|---|---|---|
| $3$ | 3.1.204.1 | \(\Z/2\Z\) | not in database |
| $4$ | 4.4.122825.1 | \(\Z/5\Z\) | not in database |
| $6$ | 6.0.2122416.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/10\Z\) | not in database |
| $20$ | 20.0.3760312697824452795491478057861328125.1 | \(\Z/5\Z\) | 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 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | ss | split | add | ord | ord | ord | add | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4,19 | 3 | - | 0 | 0 | 2 | - | 0 | 2 | 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 $0$.