Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3-x^2-450042x-106199884\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3-x^2z-450042xz^2-106199884z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-7200675x-6803993250\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-467, 1633)$ | $5.0714717860734505349264925771$ | $\infty$ |
| $(-476, 238)$ | $0$ | $2$ |
Integral points
\( \left(-476, 238\right) \), \( \left(-467, 1633\right) \), \( \left(-467, -1166\right) \), \( \left(25124, 3968238\right) \), \( \left(25124, -3993362\right) \)
Invariants
| Conductor: | $N$ | = | \( 40950 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 13$ |
|
| Discriminant: | $\Delta$ | = | $951035904000000000$ | = | $2^{20} \cdot 3^{6} \cdot 5^{9} \cdot 7^{2} \cdot 13 $ |
|
| j-invariant: | $j$ | = | \( \frac{884984855328729}{83492864000} \) | = | $2^{-20} \cdot 3^{3} \cdot 5^{-3} \cdot 7^{-2} \cdot 13^{-1} \cdot 32003^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1878422178310077794930670453$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.83381711727990274649506476023$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9692185652309827$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.770657435829066$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.0714717860734505349264925771$ |
|
| Real period: | $\Omega$ | ≈ | $0.18544119310874850961269546419$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot2\cdot2\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $3.7618391153072658044063176842 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 3.761839115 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.185441 \cdot 5.071472 \cdot 16}{2^2} \\ & \approx 3.761839115\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 737280 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{20}$ | nonsplit multiplicative | 1 | 1 | 20 | 20 |
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $2$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 3 |
| $7$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $13$ | $1$ | $I_{1}$ | split 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 \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 1243 & 198 \\ 330 & 67 \end{array}\right),\left(\begin{array}{rr} 1553 & 8 \\ 1552 & 9 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 1554 & 1555 \end{array}\right),\left(\begin{array}{rr} 932 & 519 \\ 1329 & 1034 \end{array}\right),\left(\begin{array}{rr} 73 & 588 \\ 1110 & 847 \end{array}\right),\left(\begin{array}{rr} 328 & 3 \\ 1245 & 1042 \end{array}\right),\left(\begin{array}{rr} 1039 & 0 \\ 0 & 1559 \end{array}\right)$.
The torsion field $K:=\Q(E[1560])$ is a degree-$19322634240$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1560\Z)$.
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 | $4$ | \( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \) |
| $3$ | additive | $6$ | \( 4550 = 2 \cdot 5^{2} \cdot 7 \cdot 13 \) |
| $5$ | additive | $18$ | \( 819 = 3^{2} \cdot 7 \cdot 13 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 40950bg
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 910a1, its twist by $-15$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{65}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-195}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{65})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.876096000000.17 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | nonsplit | add | add | nonsplit | ord | split | ord | ord | ord | ord | ord | ord | ord | ss | ord |
| $\lambda$-invariant(s) | 5 | - | - | 1 | 1 | 4 | 1 | 1 | 3 | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.