Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3-x^2+1206x+16308\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3-x^2z+1206xz^2+16308z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+19293x+1063006\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(12, 174\right) \) | $0.18063395373580054409348646238$ | $\infty$ |
| \( \left(-12, 6\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([12:174:1]\) | $0.18063395373580054409348646238$ | $\infty$ |
| \([-12:6:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(47, 1440\right) \) | $0.18063395373580054409348646238$ | $\infty$ |
| \( \left(-49, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-12, 6\right) \), \( \left(-3, 114\right) \), \( \left(-3, -111\right) \), \( \left(12, 174\right) \), \( \left(12, -186\right) \), \( \left(37, 314\right) \), \( \left(37, -351\right) \), \( \left(57, 489\right) \), \( \left(57, -546\right) \), \( \left(172, 2214\right) \), \( \left(172, -2386\right) \), \( \left(372, 7014\right) \), \( \left(372, -7386\right) \), \( \left(1713, 70041\right) \), \( \left(1713, -71754\right) \)
\([-12:6:1]\), \([-3:114:1]\), \([-3:-111:1]\), \([12:174:1]\), \([12:-186:1]\), \([37:314:1]\), \([37:-351:1]\), \([57:489:1]\), \([57:-546:1]\), \([172:2214:1]\), \([172:-2386:1]\), \([372:7014:1]\), \([372:-7386:1]\), \([1713:70041:1]\), \([1713:-71754:1]\)
\( \left(-49, 0\right) \), \((-13,\pm 900)\), \((47,\pm 1440)\), \((147,\pm 2660)\), \((227,\pm 4140)\), \((687,\pm 18400)\), \((1487,\pm 57600)\), \((6851,\pm 567180)\)
Invariants
| Conductor: | $N$ | = | \( 2070 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 23$ |
|
| Minimal Discriminant: | $\Delta$ | = | $-231384600000$ | = | $-1 \cdot 2^{6} \cdot 3^{7} \cdot 5^{5} \cdot 23^{2} $ |
|
| j-invariant: | $j$ | = | \( \frac{265971760991}{317400000} \) | = | $2^{-6} \cdot 3^{-1} \cdot 5^{-5} \cdot 23^{-2} \cdot 59^{3} \cdot 109^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.86683499728619552407695005483$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.31752885295214067837932743637$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0183573829896453$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.3166064462244975$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.18063395373580054409348646238$ |
|
| Real period: | $\Omega$ | ≈ | $0.66316010740757465029021832075$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 80 $ = $ 2\cdot2^{2}\cdot5\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $2.3957846432177671815220616470 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 2.395784643 \approx L'(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.663160 \cdot 0.180634 \cdot 80}{2^2} \\ & \approx 2.395784643\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1920 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $3$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $5$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $23$ | $2$ | $I_{2}$ | nonsplit 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 2757 & 4 \\ 2756 & 5 \end{array}\right),\left(\begin{array}{rr} 922 & 1 \\ 919 & 0 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1381 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 1201 & 4 \\ 2402 & 9 \end{array}\right),\left(\begin{array}{rr} 554 & 1 \\ 1103 & 0 \end{array}\right),\left(\begin{array}{rr} 1729 & 1036 \\ 344 & 2415 \end{array}\right)$.
The torsion field $K:=\Q(E[2760])$ is a degree-$787910492160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2760\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 45 = 3^{2} \cdot 5 \) |
| $3$ | additive | $8$ | \( 115 = 5 \cdot 23 \) |
| $5$ | split multiplicative | $6$ | \( 414 = 2 \cdot 3^{2} \cdot 23 \) |
| $23$ | nonsplit multiplicative | $24$ | \( 90 = 2 \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 2070g
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 690i1, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-13 +2 \sqrt{46}})\) | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.58027829760000.10 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.6170256000000.27 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.30983121016875.4 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | 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 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | add | split | ss | ord | ss | ord | ord | nonsplit | ss | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 3 | - | 2 | 1,1 | 1 | 1,1 | 1 | 1 | 1 | 1,1 | 3 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0 | - | 0 | 0,0 | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.