Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-544x+13888\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-544xz^2+13888z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-44091x+9992106\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(32, 168\right) \) | $0.74629920869451370890943760363$ | $\infty$ |
| \( \left(18, 98\right) \) | $0$ | $4$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([32:168:1]\) | $0.74629920869451370890943760363$ | $\infty$ |
| \([18:98:1]\) | $0$ | $4$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(285, 4536\right) \) | $0.74629920869451370890943760363$ | $\infty$ |
| \( \left(159, 2646\right) \) | $0$ | $4$ |
Integral points
\( \left(-31, 0\right) \), \((-24,\pm 112)\), \((-6,\pm 130)\), \((18,\pm 98)\), \((32,\pm 168)\), \((312,\pm 5488)\)
\([-31:0:1]\), \([-24:\pm 112:1]\), \([-6:\pm 130:1]\), \([18:\pm 98:1]\), \([32:\pm 168:1]\), \([312:\pm 5488:1]\)
\( \left(-31, 0\right) \), \((-24,\pm 112)\), \((-6,\pm 130)\), \((18,\pm 98)\), \((32,\pm 168)\), \((312,\pm 5488)\)
Invariants
| Conductor: | $N$ | = | \( 336 \) | = | $2^{4} \cdot 3 \cdot 7$ |
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| Minimal Discriminant: | $\Delta$ | = | $-70837874688$ | = | $-1 \cdot 2^{12} \cdot 3 \cdot 7^{8} $ |
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| j-invariant: | $j$ | = | \( -\frac{4354703137}{17294403} \) | = | $-1 \cdot 3^{-1} \cdot 7^{-8} \cdot 23^{3} \cdot 71^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.76735256395088508017549797613$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.074205383390939770758265854672$ |
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| $abc$ quality: | $Q$ | ≈ | $1.042661241253942$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.599757174355421$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.74629920869451370890943760363$ |
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| Real period: | $\Omega$ | ≈ | $0.95549489037591459827657410938$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot1\cdot2^{3} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.4261701611983923742568767354 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.426170161 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.955495 \cdot 0.746299 \cdot 32}{4^2} \\ & \approx 1.426170161\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 256 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}^{*}$ | additive | -1 | 4 | 12 | 0 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $7$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 8.48.0.161 | $48$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 336 = 2^{4} \cdot 3 \cdot 7 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 5 & 4 \\ 332 & 333 \end{array}\right),\left(\begin{array}{rr} 241 & 16 \\ 248 & 129 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 232 & 1 \\ 191 & 10 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 238 & 323 \end{array}\right),\left(\begin{array}{rr} 281 & 320 \\ 282 & 263 \end{array}\right),\left(\begin{array}{rr} 321 & 16 \\ 320 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 13 & 16 \\ 164 & 105 \end{array}\right)$.
The torsion field $K:=\Q(E[336])$ is a degree-$12386304$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/336\Z)$.
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 | $2$ | \( 3 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 112 = 2^{4} \cdot 7 \) |
| $7$ | split multiplicative | $8$ | \( 48 = 2^{4} \cdot 3 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 336.a
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 21.a4, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | 2.0.3.1-37632.2-f2 |
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/8\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-6}) \) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{-3})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.2985984.1 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.47775744.1 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.4.362538860544.5 | \(\Z/16\Z\) | not in database |
| $8$ | 8.0.7341411926016.73 | \(\Z/16\Z\) | not in database |
| $8$ | 8.2.108884466432.1 | \(\Z/12\Z\) | not in database |
| $16$ | 16.0.36520347436056576.1 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | 16.0.328683126924509184.2 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | 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 | split | ord | ord | ord | ord | ss | ord | ss | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1,1 | 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 | 0,0 |
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.