Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3+40425x+3172750\)
|
(homogenize, simplify) |
|
\(y^2z=x^3+40425xz^2+3172750z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+40425x+3172750\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-70, 0)$ | $0$ | $2$ |
Integral points
\( \left(-70, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 44100 \) | = | $2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ |
|
| Discriminant: | $\Delta$ | = | $-8576612100000000$ | = | $-1 \cdot 2^{8} \cdot 3^{6} \cdot 5^{8} \cdot 7^{6} $ |
|
| j-invariant: | $j$ | = | \( \frac{21296}{25} \) | = | $2^{4} \cdot 5^{-2} \cdot 11^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7436036698432143690852686369$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.0454746256088441894102314342$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.8396384887826309$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.064193944414755$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.27563079070719229110643934413$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ 3\cdot2\cdot2^{2}\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L(E,1)$ | ≈ | $3.3075694884863074932772721295 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 3.307569488 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.275631 \cdot 1.000000 \cdot 48}{2^2} \\ & \approx 3.307569488\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 207360 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
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 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \), index $384$, genus $9$, and generators
$\left(\begin{array}{rr} 741 & 224 \\ 434 & 237 \end{array}\right),\left(\begin{array}{rr} 99 & 364 \\ 791 & 785 \end{array}\right),\left(\begin{array}{rr} 825 & 98 \\ 826 & 517 \end{array}\right),\left(\begin{array}{rr} 421 & 504 \\ 126 & 505 \end{array}\right),\left(\begin{array}{rr} 817 & 24 \\ 816 & 25 \end{array}\right),\left(\begin{array}{rr} 599 & 0 \\ 0 & 839 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 16 \\ 314 & 335 \end{array}\right),\left(\begin{array}{rr} 13 & 24 \\ 132 & 373 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[840])$ is a degree-$185794560$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/840\Z)$.
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 | $2$ | \( 11025 = 3^{2} \cdot 5^{2} \cdot 7^{2} \) |
| $3$ | additive | $2$ | \( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \) |
| $5$ | additive | $18$ | \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
| $7$ | additive | $26$ | \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 44100bg
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 20a1, its twist by $105$.
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 |
| $2$ | \(\Q(\sqrt{105}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.2.176400.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{105})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.154350000.1 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.497871360000.50 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.31863767040000.219 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.31116960000.9 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.6.2110564911175266105000000000000.1 | \(\Z/18\Z\) | 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 | 5 | 7 |
|---|---|---|---|---|
| Reduction type | add | add | add | add |
| $\lambda$-invariant(s) | - | - | - | - |
| $\mu$-invariant(s) | - | - | - | - |
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 $0$.