Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-53x-131\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-53xz^2-131z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-4320x-108432\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-4, 1)$ | $1.0256771628213823009669833251$ | $\infty$ |
Integral points
\((-4,\pm 1)\)
Invariants
| Conductor: | $N$ | = | \( 592 \) | = | $2^{4} \cdot 37$ |
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| Discriminant: | $\Delta$ | = | $151552$ | = | $2^{12} \cdot 37 $ |
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| j-invariant: | $j$ | = | \( \frac{4096000}{37} \) | = | $2^{15} \cdot 5^{3} \cdot 37^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.18338354858446102697122810994$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.87653072914440633638846023140$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8826782160855301$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.68814336054162$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.0256771628213823009669833251$ |
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| Real period: | $\Omega$ | ≈ | $1.7676106702337894758813231445$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.8129978972181951856578445525 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.812997897 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.767611 \cdot 1.025677 \cdot 1}{1^2} \\ & \approx 1.812997897\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 48 |
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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 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II^{*}$ | additive | -1 | 4 | 12 | 0 |
| $37$ | $1$ | $I_{1}$ | 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 |
|---|---|---|
| $3$ | 3B | 27.36.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3996 = 2^{2} \cdot 3^{3} \cdot 37 \), index $1296$, genus $43$, and generators
$\left(\begin{array}{rr} 3434 & 3987 \\ 2983 & 3206 \end{array}\right),\left(\begin{array}{rr} 1997 & 0 \\ 0 & 3995 \end{array}\right),\left(\begin{array}{rr} 257 & 1275 \\ 973 & 1312 \end{array}\right),\left(\begin{array}{rr} 1 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 54 & 1 \end{array}\right),\left(\begin{array}{rr} 31 & 36 \\ 2170 & 1231 \end{array}\right),\left(\begin{array}{rr} 2051 & 3942 \\ 2052 & 3941 \end{array}\right),\left(\begin{array}{rr} 3943 & 54 \\ 3942 & 55 \end{array}\right),\left(\begin{array}{rr} 28 & 27 \\ 729 & 703 \end{array}\right)$.
The torsion field $K:=\Q(E[3996])$ is a degree-$42507721728$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3996\Z)$.
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 | $2$ | \( 37 \) |
| $37$ | split multiplicative | $38$ | \( 16 = 2^{4} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 592.a
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 37.b3, 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/3\Z\) | 2.0.4.1-1369.2-b2 |
| $3$ | 3.3.148.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.6.810448.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.3238550208.6 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.119946304.1 | \(\Z/9\Z\) | not in database |
| $6$ | 6.0.350464.1 | \(\Z/6\Z\) | 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 |
| $12$ | 12.0.5596214759424.1 | \(\Z/9\Z\) | not in database |
| $12$ | 12.0.168147445940224.2 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.6.2976016435590060807347982714273792.1 | \(\Z/6\Z\) | not in database |
| $18$ | 18.0.151197298968148189165675085824.1 | \(\Z/18\Z\) | 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 | ord | ord | ord | ord | split | ord | ord | ord |
| $\lambda$-invariant(s) | - | 3 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.