Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-685461x+218435006\)
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(homogenize, simplify) |
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\(y^2z=x^3-685461xz^2+218435006z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-685461x+218435006\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(478, 0)$ | $0$ | $2$ |
Integral points
\( \left(478, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 14976 \) | = | $2^{7} \cdot 3^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $2665084032$ | = | $2^{7} \cdot 3^{6} \cdot 13^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{381710814819113696}{28561} \) | = | $2^{5} \cdot 7^{6} \cdot 13^{-4} \cdot 4663^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7040112032533418295820777602$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.75036920359265222005773640422$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1130169758023178$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.401085998403531$ | |||
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: | $\Omega$ | ≈ | $0.79812478726659133651120770910$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 1\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.79812478726659133651120770910 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.798124787 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.798125 \cdot 1.000000 \cdot 4}{2^2} \\ & \approx 0.798124787\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 107520 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II$ | additive | 1 | 7 | 7 | 0 |
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $13$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
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 | 16.24.0.28 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1248 = 2^{5} \cdot 3 \cdot 13 \), index $96$, genus $0$, and generators
$\left(\begin{array}{rr} 577 & 864 \\ 960 & 991 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1241 & 8 \\ 1240 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 460 & 1071 \\ 237 & 934 \end{array}\right),\left(\begin{array}{rr} 415 & 0 \\ 0 & 1247 \end{array}\right),\left(\begin{array}{rr} 253 & 420 \\ 36 & 709 \end{array}\right),\left(\begin{array}{rr} 7 & 8 \\ 20 & 23 \end{array}\right)$.
The torsion field $K:=\Q(E[1248])$ is a degree-$5152702464$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1248\Z)$.
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 | $4$ | \( 9 = 3^{2} \) |
| $3$ | additive | $2$ | \( 1664 = 2^{7} \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 1152 = 2^{7} \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 14976.g
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 1664.d1, its twist by $-24$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.4608.1 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.339738624.8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.5435817984.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.3234424946688.27 | \(\Z/6\Z\) | not in database |
| $16$ | 16.0.118192468620711297024.1 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | 16.0.472769874482845188096.29 | \(\Z/8\Z\) | 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 | 13 |
|---|---|---|---|
| Reduction type | add | add | nonsplit |
| $\lambda$-invariant(s) | - | - | 0 |
| $\mu$-invariant(s) | - | - | 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 $0$.