Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-738784x+144711424\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-738784xz^2+144711424z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-59841531x+105315103530\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(208, 0)$ | $0$ | $2$ |
| $(1680, 60352)$ | $0$ | $4$ |
Integral points
\( \left(-944, 0\right) \), \((-206,\pm 16974)\), \( \left(208, 0\right) \), \( \left(737, 0\right) \), \((1680,\pm 60352)\)
Invariants
| Conductor: | $N$ | = | \( 45264 \) | = | $2^{4} \cdot 3 \cdot 23 \cdot 41$ |
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| Discriminant: | $\Delta$ | = | $16790812492845809664$ | = | $2^{18} \cdot 3^{4} \cdot 23^{4} \cdot 41^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{10887214909148618977}{4099319456261184} \) | = | $2^{-6} \cdot 3^{-4} \cdot 23^{-4} \cdot 41^{-4} \cdot 73^{3} \cdot 97^{3} \cdot 313^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3885548891619276275796825023$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.6954077086019823181624503808$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9818715313962838$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.864793532593644$ | |||
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: | $\Omega$ | ≈ | $0.20041530204419104433176429058$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2^{2}\cdot2\cdot2^{2}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $8$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.6033224163535283546541143246 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 1.603322416 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.200415 \cdot 1.000000 \cdot 128}{8^2} \\ & \approx 1.603322416\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 958464 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{10}^{*}$ | additive | -1 | 4 | 18 | 6 |
| $3$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $23$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $41$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
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 |
|---|---|---|
| $2$ | 2Cs | 8.48.0.23 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 7544 = 2^{3} \cdot 23 \cdot 41 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 5 & 4 \\ 7540 & 7541 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 5249 & 8 \\ 5908 & 33 \end{array}\right),\left(\begin{array}{rr} 7537 & 1880 \\ 7522 & 5639 \end{array}\right),\left(\begin{array}{rr} 3 & 1890 \\ 1882 & 7539 \end{array}\right),\left(\begin{array}{rr} 7537 & 8 \\ 7536 & 9 \end{array}\right),\left(\begin{array}{rr} 5705 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[7544])$ is a degree-$5888810188800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/7544\Z)$.
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 | $2$ | \( 1 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 15088 = 2^{4} \cdot 23 \cdot 41 \) |
| $23$ | split multiplicative | $24$ | \( 1968 = 2^{4} \cdot 3 \cdot 41 \) |
| $41$ | split multiplicative | $42$ | \( 1104 = 2^{4} \cdot 3 \cdot 23 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 45264l
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 5658g3, its twist by $-4$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(\zeta_{8})\) | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-46}, \sqrt{82})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{23}, \sqrt{41})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | 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 | 23 | 41 |
|---|---|---|---|---|
| Reduction type | add | nonsplit | split | split |
| $\lambda$-invariant(s) | - | 0 | 1 | 1 |
| $\mu$-invariant(s) | - | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ 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 $0$.