Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-493968x-1075051756\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-493968xz^2-1075051756z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-40011435x-783592695846\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{25942738}{8649}, \frac{125744906240}{804357}\right) \) | $8.2732441234299943860515444307$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([2412674634:125744906240:804357]\) | $8.2732441234299943860515444307$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{25945621}{961}, \frac{125744906240}{29791}\right) \) | $8.2732441234299943860515444307$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 5776 \) | = | $2^{4} \cdot 19^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-491411185385426911232$ | = | $-1 \cdot 2^{39} \cdot 19^{7} $ |
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| j-invariant: | $j$ | = | \( -\frac{69173457625}{2550136832} \) | = | $-1 \cdot 2^{-27} \cdot 5^{3} \cdot 19^{-1} \cdot 821^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.6505362811004091203737077575$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.48516961095724358095196192010$ |
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| $abc$ quality: | $Q$ | ≈ | $1.054621266416818$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.363138081521333$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $8.2732441234299943860515444307$ |
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| Real period: | $\Omega$ | ≈ | $0.072290130085258183194926304522$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.7845911512788168252694262883 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.784591151 \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 0.072290 \cdot 8.273244 \cdot 8}{1^2} \\ & \approx 4.784591151\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 155520 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{31}^{*}$ | additive | -1 | 4 | 39 | 27 |
| $19$ | $2$ | $I_{1}^{*}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $3$ | 3B | 27.36.0.1 | $36$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4104 = 2^{3} \cdot 3^{3} \cdot 19 \), index $1296$, genus $43$, and generators
$\left(\begin{array}{rr} 937 & 75 \\ 1653 & 2144 \end{array}\right),\left(\begin{array}{rr} 31 & 36 \\ 2386 & 1447 \end{array}\right),\left(\begin{array}{rr} 1931 & 4068 \\ 397 & 3575 \end{array}\right),\left(\begin{array}{rr} 1 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2053 & 54 \\ 2061 & 487 \end{array}\right),\left(\begin{array}{rr} 1025 & 4050 \\ 0 & 4103 \end{array}\right),\left(\begin{array}{rr} 28 & 27 \\ 729 & 703 \end{array}\right),\left(\begin{array}{rr} 4051 & 54 \\ 4050 & 55 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 54 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[4104])$ is a degree-$45954293760$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4104\Z)$.
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 | $4$ | \( 361 = 19^{2} \) |
| $19$ | additive | $200$ | \( 16 = 2^{4} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 5776l
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 38a2, its twist by $76$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-57}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.152.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.3511808.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.1426233024.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.3119171623488.5 | \(\Z/9\Z\) | not in database |
| $6$ | 6.0.189637632.1 | \(\Z/6\Z\) | not in database |
| $12$ | 12.2.119973433931988992.13 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | 12.0.26950780101863616712704.3 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.2.11883145438998998476139060527104.2 | \(\Z/6\Z\) | not in database |
| $18$ | 18.0.124301898454548014823673340858103194714112.1 | \(\Z/18\Z\) | 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 | add | ord | ss | ord | ord | ord | ord | add | ord | ord | ord | ord | ss | ord | ss |
| $\lambda$-invariant(s) | - | 3 | 1,1 | 1 | 1 | 1 | 1 | - | 1 | 1 | 1 | 1 | 1,1 | 1 | 1,1 |
| $\mu$-invariant(s) | - | 2 | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.