Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-1464x+20596\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-1464xz^2+20596z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-118611x+15370290\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(26, 28)$ | $0.23398755000200838043894506435$ | $\infty$ |
| $(18, 20)$ | $1.7672914374876096082409015182$ | $\infty$ |
| $(19, 0)$ | $0$ | $2$ |
Integral points
\((-44,\pm 42)\), \((-30,\pm 196)\), \((-9,\pm 182)\), \((10,\pm 84)\), \((12,\pm 70)\), \((18,\pm 20)\), \( \left(19, 0\right) \), \((26,\pm 28)\), \((35,\pm 116)\), \((68,\pm 490)\), \((140,\pm 1606)\), \((194,\pm 2660)\), \((362,\pm 6860)\), \((5906,\pm 453908)\)
Invariants
| Conductor: | $N$ | = | \( 4144 \) | = | $2^{4} \cdot 7 \cdot 37$ |
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| Discriminant: | $\Delta$ | = | $8914970624$ | = | $2^{11} \cdot 7^{6} \cdot 37 $ |
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| j-invariant: | $j$ | = | \( \frac{169556172914}{4353013} \) | = | $2 \cdot 7^{-6} \cdot 23^{3} \cdot 37^{-1} \cdot 191^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.69149075613483605871100971300$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.056105840621552858411880268330$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8893097731815562$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.019617455474731$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.40934235608067877257130465381$ |
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| Real period: | $\Omega$ | ≈ | $1.2979831696735584785651763883$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 24 $ = $ 2^{2}\cdot( 2 \cdot 3 )\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $3.1879169329634512048027138000 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.187916933 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.297983 \cdot 0.409342 \cdot 24}{2^2} \\ & \approx 3.187916933\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3456 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{3}^{*}$ | additive | 1 | 4 | 11 | 0 |
| $7$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $37$ | $1$ | $I_{1}$ | 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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2072 = 2^{3} \cdot 7 \cdot 37 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 1035 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 297 & 4 \\ 594 & 9 \end{array}\right),\left(\begin{array}{rr} 226 & 1 \\ 1959 & 0 \end{array}\right),\left(\begin{array}{rr} 1297 & 778 \\ 776 & 1295 \end{array}\right),\left(\begin{array}{rr} 2069 & 4 \\ 2068 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right)$.
The torsion field $K:=\Q(E[2072])$ is a degree-$470208872448$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2072\Z)$.
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 | $4$ | \( 37 \) |
| $3$ | good | $2$ | \( 592 = 2^{4} \cdot 37 \) |
| $7$ | split multiplicative | $8$ | \( 592 = 2^{4} \cdot 37 \) |
| $37$ | nonsplit multiplicative | $38$ | \( 112 = 2^{4} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 4144e
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 2072c2, its twist by $-4$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{74}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.14504.1 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.18431428857856.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.16788644278272.19 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | ord | ord | split | ord | ord | ord | ord | ss | ord | ord | nonsplit | ord | ord | ss |
| $\lambda$-invariant(s) | - | 2 | 2 | 3 | 2 | 2 | 2 | 2 | 2,4 | 2 | 2 | 2 | 2 | 2 | 2,2 |
| $\mu$-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.