Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-64952x-9118032\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-64952xz^2-9118032z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-5261139x-6662828718\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(308, 0)$ | $0$ | $2$ |
Integral points
\( \left(308, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 89232 \) | = | $2^{4} \cdot 3 \cdot 11 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $-18549240710787072$ | = | $-1 \cdot 2^{12} \cdot 3^{8} \cdot 11 \cdot 13^{7} $ |
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| j-invariant: | $j$ | = | \( -\frac{1532808577}{938223} \) | = | $-1 \cdot 3^{-8} \cdot 11^{-1} \cdot 13^{-1} \cdot 1153^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.8235703176376831893965013360$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.15205154165303048804747450624$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8840474094399083$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.998547065472629$ | |||
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: | $\Omega$ | ≈ | $0.14547327974974701240507135653$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2^{2}\cdot2\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.58189311899898804962028542612 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.581893119 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.145473 \cdot 1.000000 \cdot 16}{2^2} \\ & \approx 0.581893119\end{aligned}$$
Modular invariants
Modular form 89232.2.a.z
For more coefficients, see the Downloads section to the right.
| Modular degree: | 688128 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}^{*}$ | additive | -1 | 4 | 12 | 0 |
| $3$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $11$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $13$ | $2$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 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 | 8.24.0.93 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 5792 & 6859 \\ 2157 & 14 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 1724 & 1845 \end{array}\right),\left(\begin{array}{rr} 4577 & 16 \\ 2296 & 129 \end{array}\right),\left(\begin{array}{rr} 6849 & 16 \\ 6848 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 6766 & 6851 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 6860 & 6861 \end{array}\right),\left(\begin{array}{rr} 632 & 1 \\ 1951 & 10 \end{array}\right),\left(\begin{array}{rr} 847 & 6848 \\ 2374 & 1425 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[6864])$ is a degree-$2125489766400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6864\Z)$.
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 | $2$ | \( 1859 = 11 \cdot 13^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 29744 = 2^{4} \cdot 11 \cdot 13^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 528 = 2^{4} \cdot 3 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 89232bd
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 429b1, its twist by $-52$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-143}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{11}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-13}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-11}, \sqrt{-13})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{13})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{11}, \sqrt{-13})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.136815785261584.2 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.2189052564185344.1 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.107049369856.1 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | 11 | 13 |
|---|---|---|---|---|
| Reduction type | add | nonsplit | nonsplit | add |
| $\lambda$-invariant(s) | - | 0 | 0 | - |
| $\mu$-invariant(s) | - | 0 | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.