Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-50342240x-138322037709\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-50342240xz^2-138322037709z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-805475835x-8853415889194\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(32795/4, -32799/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 45486 \) | = | $2 \cdot 3^{2} \cdot 7 \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $-101552129458604613320112$ | = | $-1 \cdot 2^{4} \cdot 3^{3} \cdot 7^{12} \cdot 19^{8} $ |
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| j-invariant: | $j$ | = | \( -\frac{11108001800138902875}{79947274872976} \) | = | $-1 \cdot 2^{-4} \cdot 3^{6} \cdot 5^{3} \cdot 7^{-12} \cdot 19^{-2} \cdot 179^{3} \cdot 277^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $3.2468371450142924221989085666$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.4999645832640447693455835414$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0300193263247077$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.044574546067864$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.028326944412958632884232216196$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 384 $ = $ 2^{2}\cdot2\cdot( 2^{2} \cdot 3 )\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.7193866636440287568862927548 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 2.719386664 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.028327 \cdot 1.000000 \cdot 384}{2^2} \\ & \approx 2.719386664\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6635520 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $7$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
| $19$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 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 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 228 = 2^{2} \cdot 3 \cdot 19 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 11 & 2 \\ 178 & 219 \end{array}\right),\left(\begin{array}{rr} 219 & 136 \\ 182 & 125 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 131 & 216 \\ 102 & 155 \end{array}\right),\left(\begin{array}{rr} 164 & 11 \\ 117 & 196 \end{array}\right),\left(\begin{array}{rr} 217 & 12 \\ 216 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[228])$ is a degree-$5909760$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/228\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 1083 = 3 \cdot 19^{2} \) |
| $3$ | additive | $6$ | \( 722 = 2 \cdot 19^{2} \) |
| $7$ | split multiplicative | $8$ | \( 6498 = 2 \cdot 3^{2} \cdot 19^{2} \) |
| $19$ | additive | $200$ | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 45486.bf
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 2394.c3, its twist by $-19$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-19}) \) | \(\Z/6\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.2.155952.2 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{-19})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.86643656208.2 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.24321026304.12 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.24321026304.2 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | 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 |
| $18$ | 18.0.30679685227924636315843364396486718810237696.3 | \(\Z/18\Z\) | 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 | 7 | 19 |
|---|---|---|---|---|
| Reduction type | split | add | split | add |
| $\lambda$-invariant(s) | 4 | - | 1 | - |
| $\mu$-invariant(s) | 1 | - | 0 | - |
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 $0$.