Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-42123x+3327482\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-42123xz^2+3327482z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-42123x+3327482\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(79, 702)$ | $1.3197736812422734108701236103$ | $\infty$ |
| $(119, 2)$ | $2.2412641842559404732625671182$ | $\infty$ |
| $(118, 0)$ | $0$ | $2$ |
Integral points
\((-233,\pm 702)\), \((79,\pm 702)\), \( \left(118, 0\right) \), \((119,\pm 2)\), \((127,\pm 162)\), \((154,\pm 702)\), \((743,\pm 19550)\), \((7862,\pm 696872)\)
Invariants
| Conductor: | $N$ | = | \( 102960 \) | = | $2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 13$ |
|
| Discriminant: | $\Delta$ | = | $249792491520$ | = | $2^{12} \cdot 3^{8} \cdot 5 \cdot 11 \cdot 13^{2} $ |
|
| j-invariant: | $j$ | = | \( \frac{2768178670921}{83655} \) | = | $3^{-2} \cdot 5^{-1} \cdot 11^{-1} \cdot 13^{-2} \cdot 19^{3} \cdot 739^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2851683611062063031893047812$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.042715036212206148074450041281$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.8886684168571962$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.7738943770982547$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
|
| Mordell-Weil rank: | $r$ | = | $ 2$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.7848209929027886054603395549$ |
|
| Real period: | $\Omega$ | ≈ | $0.91834616683334749444058548754$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2^{2}\cdot1\cdot1\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $10.229718736597250893736287520 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 10.229718737 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.918346 \cdot 2.784821 \cdot 16}{2^2} \\ & \approx 10.229718737\end{aligned}$$
Modular invariants
Modular form 102960.2.a.p
For more coefficients, see the Downloads section to the right.
| Modular degree: | 229376 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{4}^{*}$ | additive | -1 | 4 | 12 | 0 |
| $3$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $11$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $13$ | $2$ | $I_{2}$ | 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 | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 8580 = 2^{2} \cdot 3 \cdot 5 \cdot 11 \cdot 13 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 2641 & 4 \\ 5282 & 9 \end{array}\right),\left(\begin{array}{rr} 2861 & 4 \\ 5722 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7022 & 1 \\ 779 & 0 \end{array}\right),\left(\begin{array}{rr} 3434 & 1 \\ 6863 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 6436 & 2149 \\ 2145 & 6436 \end{array}\right),\left(\begin{array}{rr} 8577 & 4 \\ 8576 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[8580])$ is a degree-$63764692992000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8580\Z)$.
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 | $2$ | \( 495 = 3^{2} \cdot 5 \cdot 11 \) |
| $3$ | additive | $8$ | \( 11440 = 2^{4} \cdot 5 \cdot 11 \cdot 13 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 20592 = 2^{4} \cdot 3^{2} \cdot 11 \cdot 13 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 102960dg
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 2145f2, its twist by $12$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{55}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.334620.4 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.5419374348960000.9 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | 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 | add | add | nonsplit | ord | nonsplit | split | ord | ord | ord | ss | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | 4 | 2 | 2 | 5 | 4 | 2 | 2 | 2,2 | 2 | 2 | 2 | 2 | 2 |
| $\mu$-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.