Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2+16285x+1020323\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z+16285xz^2+1020323z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+260565x+65561254\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{6}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(291, 5362)$ | $0$ | $6$ |
Integral points
\( \left(19, 1146\right) \), \( \left(19, -1166\right) \), \( \left(291, 5362\right) \), \( \left(291, -5654\right) \)
Invariants
| Conductor: | $N$ | = | \( 306 \) | = | $2 \cdot 3^{2} \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $-729753247683072$ | = | $-1 \cdot 2^{9} \cdot 3^{10} \cdot 17^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{655215969476375}{1001033261568} \) | = | $2^{-9} \cdot 3^{-4} \cdot 5^{3} \cdot 17^{-6} \cdot 29^{3} \cdot 599^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.5373060552954471549544165371$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.98799991096139230925679391864$ |
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| $abc$ quality: | $Q$ | ≈ | $1.053583581360511$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.198699681285715$ | |||
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: | $\Omega$ | ≈ | $0.34471767962369798629289706239$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 216 $ = $ 3^{2}\cdot2^{2}\cdot( 2 \cdot 3 ) $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $6$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.0683060777421879177573823744 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 2.068306078 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.344718 \cdot 1.000000 \cdot 216}{6^2} \\ & \approx 2.068306078\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1152 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $9$ | $I_{9}$ | split multiplicative | -1 | 1 | 9 | 9 |
| $3$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $17$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
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 | 8.6.0.5 |
| $3$ | 3B.1.1 | 3.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 408 = 2^{3} \cdot 3 \cdot 17 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 10 & 3 \\ 177 & 400 \end{array}\right),\left(\begin{array}{rr} 241 & 12 \\ 222 & 73 \end{array}\right),\left(\begin{array}{rr} 397 & 12 \\ 396 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 18 & 97 \\ 221 & 290 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 358 & 399 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 271 & 396 \\ 68 & 407 \end{array}\right)$.
The torsion field $K:=\Q(E[408])$ is a degree-$60162048$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/408\Z)$.
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 | $4$ | \( 9 = 3^{2} \) |
| $3$ | additive | $8$ | \( 1 \) |
| $17$ | split multiplicative | $18$ | \( 18 = 2 \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 306a
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 102c4, its twist by $-3$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | 2.0.8.1-46818.8-s6 |
| $4$ | 4.2.83232.1 | \(\Z/12\Z\) | not in database |
| $6$ | 6.0.177147.2 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.0.98184462336.24 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | 8.0.443364212736.18 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $9$ | 9.3.346347732787083.4 | \(\Z/18\Z\) | not in database |
| $12$ | 12.0.8226356490141696.17 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $18$ | 18.0.16100322712605781141130566495415304192.2 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 | 17 |
|---|---|---|---|
| Reduction type | split | add | split |
| $\lambda$-invariant(s) | 2 | - | 1 |
| $\mu$-invariant(s) | 1 | - | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ 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 $0$.