Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-87767x-10014615\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-87767xz^2-10014615z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-113746059x-466900639290\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-170, 85)$ | $0$ | $2$ |
| $(342, -171)$ | $0$ | $2$ |
Integral points
\( \left(-170, 85\right) \), \( \left(342, -171\right) \)
Invariants
| Conductor: | $N$ | = | \( 1122 \) | = | $2 \cdot 3 \cdot 11 \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $5615307472896$ | = | $2^{14} \cdot 3^{4} \cdot 11^{4} \cdot 17^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{74768347616680342513}{5615307472896} \) | = | $2^{-14} \cdot 3^{-4} \cdot 7^{3} \cdot 11^{-4} \cdot 17^{-2} \cdot 601831^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.4956941239258774815133406561$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.4956941239258774815133406561$ |
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| $abc$ quality: | $Q$ | ≈ | $1.013375978983816$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.515988303951806$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.27737586688638696545624363972$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 224 $ = $ ( 2 \cdot 7 )\cdot2^{2}\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.8832621364094175163874109561 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 3.883262136 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.277376 \cdot 1.000000 \cdot 224}{4^2} \\ & \approx 3.883262136\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 5376 |
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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 semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $14$ | $I_{14}$ | split multiplicative | -1 | 1 | 14 | 14 |
| $3$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $11$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $17$ | $2$ | $I_{2}$ | split 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 |
|---|---|---|
| $2$ | 2Cs | 8.24.0.4 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 136 = 2^{3} \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 105 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 133 & 4 \\ 132 & 5 \end{array}\right),\left(\begin{array}{rr} 103 & 4 \\ 70 & 9 \end{array}\right),\left(\begin{array}{rr} 71 & 2 \\ 134 & 135 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[136])$ is a degree-$2506752$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/136\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 1 \) |
| $3$ | split multiplicative | $4$ | \( 374 = 2 \cdot 11 \cdot 17 \) |
| $7$ | good | $2$ | \( 561 = 3 \cdot 11 \cdot 17 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 102 = 2 \cdot 3 \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 66 = 2 \cdot 3 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 1122k
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
This elliptic curve is its own minimal quadratic twist.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{17})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{17})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.5473632256.1 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.98184462336.24 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.3465933379972272.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | 17 |
|---|---|---|---|---|---|---|
| Reduction type | split | split | ord | ord | nonsplit | split |
| $\lambda$-invariant(s) | 2 | 3 | 0 | 8 | 0 | 1 |
| $\mu$-invariant(s) | 1 | 0 | 0 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 11$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.