Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-1987317522x-34098857881772\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-1987317522xz^2-34098857881772z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-31797080355x-2182358701513762\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(51476, -25738)$ | $0$ | $2$ |
Integral points
\( \left(51476, -25738\right) \)
Invariants
| Conductor: | $N$ | = | \( 45486 \) | = | $2 \cdot 3^{2} \cdot 7 \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $6582637107432313011044352$ | = | $2^{36} \cdot 3^{7} \cdot 7^{2} \cdot 19^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{25309080274342544331625}{191933498523648} \) | = | $2^{-36} \cdot 3^{-1} \cdot 5^{3} \cdot 7^{-2} \cdot 19^{-1} \cdot 43^{3} \cdot 136559^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.9371852796372344271016136172$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.9156596457199593513994772828$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0385639997055625$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.071573640673038$ | |||
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: | $\Omega$ | ≈ | $0.022611685526770878975244378102$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.81402067896375164310879761169 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $9$ = $3^2$ (exact) |
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BSD formula
$$\begin{aligned} 0.814020679 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{9 \cdot 0.022612 \cdot 1.000000 \cdot 16}{2^2} \\ & \approx 0.814020679\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 29859840 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{36}$ | nonsplit multiplicative | 1 | 1 | 36 | 36 |
| $3$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $7$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $19$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
| $3$ | 3B | 9.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 \), index $864$, genus $21$, and generators
$\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 752 & 4779 \\ 1685 & 4610 \end{array}\right),\left(\begin{array}{rr} 2096 & 4761 \\ 3911 & 1654 \end{array}\right),\left(\begin{array}{rr} 2395 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2077 & 36 \\ 678 & 265 \end{array}\right),\left(\begin{array}{rr} 19 & 36 \\ 3240 & 1351 \end{array}\right),\left(\begin{array}{rr} 4753 & 36 \\ 4752 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 14 & 253 \end{array}\right)$.
The torsion field $K:=\Q(E[4788])$ is a degree-$107226685440$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4788\Z)$.
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 | $4$ | \( 3249 = 3^{2} \cdot 19^{2} \) |
| $3$ | additive | $8$ | \( 2527 = 7 \cdot 19^{2} \) |
| $7$ | split multiplicative | $8$ | \( 6498 = 2 \cdot 3^{2} \cdot 19^{2} \) |
| $19$ | additive | $200$ | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 6, 9 and 18.
Its isogeny class 45486q
consists of 6 curves linked by isogenies of
degrees dividing 18.
Twists
The minimal quadratic twist of this elliptic curve is 798e5, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{57}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-19}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.4.44688.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{-19})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.1053159056436753.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.39005890979139.9 | \(\Z/18\Z\) | not in database |
| $8$ | 8.0.26888414643216.5 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.8.6488309350656.3 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.720923261184.9 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/18\Z\) | not in database |
| $12$ | 12.0.13693135779688294945198631889.2 | \(\Z/2\Z \oplus \Z/18\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | 7 | 19 |
|---|---|---|---|---|
| Reduction type | nonsplit | add | split | add |
| $\lambda$-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 $0$.