Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-597624x-170517312\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-597624xz^2-170517312z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-48407571x-124452343134\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1428, 43428)$ | $3.7467502143731023898739824809$ | $\infty$ |
| $(-442, 2662)$ | $0$ | $4$ |
Integral points
\((-442,\pm 2662)\), \( \left(889, 0\right) \), \((1428,\pm 43428)\)
Invariants
| Conductor: | $N$ | = | \( 19536 \) | = | $2^{4} \cdot 3 \cdot 11 \cdot 37$ |
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| Discriminant: | $\Delta$ | = | $1070178969034847232$ | = | $2^{10} \cdot 3^{2} \cdot 11^{12} \cdot 37 $ |
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| j-invariant: | $j$ | = | \( \frac{23051997945147370468}{1045096649448093} \) | = | $2^{2} \cdot 3^{-2} \cdot 11^{-12} \cdot 23^{3} \cdot 37^{-1} \cdot 77951^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2216170596139264271620611234$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.6439944091473053359810343555$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9871996871198981$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.214137844352074$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.7467502143731023898739824809$ |
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| Real period: | $\Omega$ | ≈ | $0.17219054255540081534897521475$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 96 $ = $ 2^{2}\cdot2\cdot( 2^{2} \cdot 3 )\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.8709297133948128882339905830 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.870929713 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.172191 \cdot 3.746750 \cdot 96}{4^2} \\ & \approx 3.870929713\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 208896 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{2}^{*}$ | additive | 1 | 4 | 10 | 0 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $11$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
| $37$ | $1$ | $I_{1}$ | split 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 | 4.12.0.7 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3256 = 2^{3} \cdot 11 \cdot 37 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 2961 & 8 \\ 2076 & 33 \end{array}\right),\left(\begin{array}{rr} 2820 & 1 \\ 3191 & 6 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 3249 & 8 \\ 3248 & 9 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 3250 & 3251 \end{array}\right),\left(\begin{array}{rr} 1213 & 1220 \\ 1174 & 2843 \end{array}\right),\left(\begin{array}{rr} 411 & 410 \\ 1234 & 2859 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[3256])$ is a degree-$769687142400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3256\Z)$.
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 | $2$ | \( 37 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 592 = 2^{4} \cdot 37 \) |
| $11$ | split multiplicative | $12$ | \( 1776 = 2^{4} \cdot 3 \cdot 37 \) |
| $37$ | split multiplicative | $38$ | \( 528 = 2^{4} \cdot 3 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 19536g
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 9768p3, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{37}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.0.286528.4 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.13619943121158144.30 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.112392565559296.15 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.1359880186540032.1 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | 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 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | nonsplit | ord | ss | split | ord | ord | ss | ss | ord | ss | split | ord | ss | ord |
| $\lambda$-invariant(s) | - | 1 | 1 | 1,1 | 2 | 1 | 1 | 1,1 | 1,1 | 1 | 1,1 | 2 | 1 | 1,1 | 1 |
| $\mu$-invariant(s) | - | 0 | 0 | 0,0 | 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.