Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-x^2+15860x-79400\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-x^2z+15860xz^2-79400z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+1284633x-54028674\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(70, 1170)$ | $0.60156184268059707918492735059$ | $\infty$ |
| $(5, 0)$ | $0$ | $2$ |
Integral points
\( \left(5, 0\right) \), \((30,\pm 650)\), \((70,\pm 1170)\), \((250,\pm 4410)\), \((642,\pm 16562)\)
Invariants
| Conductor: | $N$ | = | \( 38220 \) | = | $2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13$ |
|
| Discriminant: | $\Delta$ | = | $-257679545760000$ | = | $-1 \cdot 2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{6} \cdot 13^{2} $ |
|
| j-invariant: | $j$ | = | \( \frac{14647977776}{8555625} \) | = | $2^{4} \cdot 3^{-4} \cdot 5^{-4} \cdot 13^{-2} \cdot 971^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.4546473136551050964275856745$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.019594118754151570930087888473$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9914250933349535$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.850608156379016$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.60156184268059707918492735059$ |
|
| Real period: | $\Omega$ | ≈ | $0.32603270314583763011880623801$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 96 $ = $ 3\cdot2\cdot2^{2}\cdot2\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $4.7070920082851084372881758106 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 4.707092008 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.326033 \cdot 0.601562 \cdot 96}{2^2} \\ & \approx 4.707092008\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 138240 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $3$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $5$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $13$ | $2$ | $I_{2}$ | split 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 |
|---|---|---|
| $2$ | 2B | 4.6.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 3 & 8 \\ 10 & 27 \end{array}\right),\left(\begin{array}{rr} 3119 & 0 \\ 0 & 3639 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 561 & 2086 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1457 & 2604 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1821 & 1568 \\ 3122 & 3137 \end{array}\right),\left(\begin{array}{rr} 2731 & 2604 \\ 847 & 1569 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 48 & 77 \end{array}\right),\left(\begin{array}{rr} 3633 & 8 \\ 3632 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[3640])$ is a degree-$811550638080$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3640\Z)$.
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 | $2$ | \( 49 = 7^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 12740 = 2^{2} \cdot 5 \cdot 7^{2} \cdot 13 \) |
| $5$ | split multiplicative | $6$ | \( 7644 = 2^{2} \cdot 3 \cdot 7^{2} \cdot 13 \) |
| $7$ | additive | $26$ | \( 780 = 2^{2} \cdot 3 \cdot 5 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 38220q
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 780c2, its twist by $-7$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.2.132496.1 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.106369908736.10 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.280883040256.9 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | 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 | add | nonsplit | split | add | ord | split | ord | ord | ord | ord | ord | ord | ord | ss | ord |
| $\lambda$-invariant(s) | - | 1 | 2 | - | 3 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1,1 | 1 |
| $\mu$-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.