Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+2220x-78536\)
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(homogenize, simplify) |
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\(y^2z=x^3+2220xz^2-78536z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+2220x-78536\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(125, 1467)$ | $3.9508835662212964503637964407$ | $\infty$ |
Integral points
\((125,\pm 1467)\)
Invariants
| Conductor: | $N$ | = | \( 131904 \) | = | $2^{6} \cdot 3^{2} \cdot 229$ |
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| Discriminant: | $\Delta$ | = | $-3364761295872$ | = | $-1 \cdot 2^{10} \cdot 3^{15} \cdot 229 $ |
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| j-invariant: | $j$ | = | \( \frac{1620896000}{4507407} \) | = | $2^{8} \cdot 3^{-9} \cdot 5^{3} \cdot 37^{3} \cdot 229^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.0872410081499454389381302594$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.039687786650730497940519126943$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8603993239697355$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.0590577946782167$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.9508835662212964503637964407$ |
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| Real period: | $\Omega$ | ≈ | $0.40786488814008708459221336683$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.2228533675827148405953115433 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.222853368 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.407865 \cdot 3.950884 \cdot 2}{1^2} \\ & \approx 3.222853368\end{aligned}$$
Modular invariants
Modular form 131904.2.a.ba
For more coefficients, see the Downloads section to the right.
| Modular degree: | 184320 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_0^{*}$ | additive | -1 | 6 | 10 | 0 |
| $3$ | $2$ | $I_{9}^{*}$ | additive | -1 | 2 | 15 | 9 |
| $229$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 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 |
|---|---|---|
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 5496 = 2^{3} \cdot 3 \cdot 229 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 4127 & 5490 \\ 4128 & 5489 \end{array}\right),\left(\begin{array}{rr} 4350 & 5261 \\ 4577 & 4332 \end{array}\right),\left(\begin{array}{rr} 4121 & 0 \\ 0 & 5495 \end{array}\right),\left(\begin{array}{rr} 235 & 6 \\ 705 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 2747 & 0 \\ 0 & 5495 \end{array}\right),\left(\begin{array}{rr} 5491 & 6 \\ 5490 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[5496])$ is a degree-$12616691466240$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5496\Z)$.
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 | $2$ | \( 2061 = 3^{2} \cdot 229 \) |
| $3$ | additive | $2$ | \( 14656 = 2^{6} \cdot 229 \) |
| $229$ | nonsplit multiplicative | $230$ | \( 576 = 2^{6} \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 131904.ba
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 2748.c2, its twist by $24$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{6}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.687.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.324242703.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.50689077921792.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.2.724944384.4 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.6.679547884397935440128740503136329793536.2 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.61469070699351108147246091214617001408197558272.1 | \(\Z/6\Z\) | 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 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 229 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | add | ss | ord | ord | ord | ord | ord | ss | ord | ord | ord | ord | ord | ord | nonsplit |
| $\lambda$-invariant(s) | - | - | 1,1 | 1 | 1 | 1 | 1 | 3 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | - | 0,0 | 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.