Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3-x^2-39317x+3010466\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3-x^2z-39317xz^2+3010466z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-629067x+192040774\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(96, 289)$ | $0.44390224053580665387572443990$ | $\infty$ |
| $(459/4, -463/8)$ | $0$ | $2$ |
Integral points
\( \left(-156, 2386\right) \), \( \left(-156, -2231\right) \), \( \left(96, 289\right) \), \( \left(96, -386\right) \), \( \left(114, -44\right) \), \( \left(114, -71\right) \), \( \left(115, -53\right) \), \( \left(115, -63\right) \), \( \left(171, 1039\right) \), \( \left(171, -1211\right) \)
Invariants
| Conductor: | $N$ | = | \( 4095 \) | = | $3^{2} \cdot 5 \cdot 7 \cdot 13$ |
|
| Discriminant: | $\Delta$ | = | $373156875$ | = | $3^{8} \cdot 5^{4} \cdot 7 \cdot 13 $ |
|
| j-invariant: | $j$ | = | \( \frac{9219915604149769}{511875} \) | = | $3^{-2} \cdot 5^{-4} \cdot 7^{-1} \cdot 13^{-1} \cdot 277^{3} \cdot 757^{3}$ |
|
| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
|
||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $1.1127864296256887987903837835$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.56348028529163395309276116504$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9536815956308902$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.2121071672830785$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.44390224053580665387572443990$ |
|
| Real period: | $\Omega$ | ≈ | $1.2742382967942691972149679167$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2^{2}\cdot2^{2}\cdot1\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $2.2625489396940250954094625866 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 2.262548940 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.274238 \cdot 0.443902 \cdot 16}{2^2} \\ & \approx 2.262548940\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 8192 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $5$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $13$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10920 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 8737 & 3648 \\ 9468 & 3673 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 7279 & 0 \\ 0 & 10919 \end{array}\right),\left(\begin{array}{rr} 5884 & 7281 \\ 1983 & 3646 \end{array}\right),\left(\begin{array}{rr} 4168 & 3 \\ 9885 & 7282 \end{array}\right),\left(\begin{array}{rr} 5008 & 10473 \\ 8673 & 5080 \end{array}\right),\left(\begin{array}{rr} 4099 & 4098 \\ 8658 & 6835 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 10914 & 10915 \end{array}\right),\left(\begin{array}{rr} 10913 & 8 \\ 10912 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[10920])$ is a degree-$38954430627840$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10920\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 819 = 3^{2} \cdot 7 \cdot 13 \) |
| $3$ | additive | $8$ | \( 455 = 5 \cdot 7 \cdot 13 \) |
| $5$ | split multiplicative | $6$ | \( 819 = 3^{2} \cdot 7 \cdot 13 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 585 = 3^{2} \cdot 5 \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 4095.d
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1365.d1, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{91}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{39}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{21}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{21}, \sqrt{39})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | 8.2.7592405385166875.8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | 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 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | ord | add | split | nonsplit | ord | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | - | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 |
| $\mu$-invariant(s) | 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.