Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2+16x+1060\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z+16xz^2+1060z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+1269x+768906\)
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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 |
|---|---|---|
| $(-6, 28)$ | $0.46797510000401676087789012871$ | $\infty$ |
| $(8, 42)$ | $0.93597236518187875060373585725$ | $\infty$ |
| $(-10, 0)$ | $0$ | $2$ |
Integral points
\( \left(-10, 0\right) \), \((-8,\pm 22)\), \((-6,\pm 28)\), \((-2,\pm 32)\), \((8,\pm 42)\), \((15,\pm 70)\), \((22,\pm 112)\), \((27,\pm 148)\), \((64,\pm 518)\), \((90,\pm 860)\), \((138,\pm 1628)\), \((232,\pm 3542)\), \((638,\pm 16128)\), \((2654,\pm 136752)\)
Invariants
| Conductor: | $N$ | = | \( 4144 \) | = | $2^{4} \cdot 7 \cdot 37$ |
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| Discriminant: | $\Delta$ | = | $-480836608$ | = | $-1 \cdot 2^{10} \cdot 7^{3} \cdot 37^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{415292}{469567} \) | = | $2^{2} \cdot 7^{-3} \cdot 37^{-2} \cdot 47^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.34491716585486340400239365227$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.23270548461175768717863311561$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9212770528271738$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.294978123222376$ | |||
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}\cdot3\cdot2 $ |
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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: | 1728 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{2}^{*}$ | additive | 1 | 4 | 10 | 0 |
| $7$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $37$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
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
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 594 & 1 \\ 1775 & 0 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1037 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1297 & 778 \\ 776 & 1295 \end{array}\right),\left(\begin{array}{rr} 2069 & 4 \\ 2068 & 5 \end{array}\right),\left(\begin{array}{rr} 113 & 4 \\ 226 & 9 \end{array}\right)$.
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 | $2$ | \( 7 \) |
| $3$ | 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 4144.a
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 2072.e2, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.2.2453248.2 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.41231739136.11 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.294902861725696.3 | \(\Z/2\Z \oplus \Z/4\Z\) | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.