Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2+3060x+64557\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z+3060xz^2+64557z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+3965733x+2952493686\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(45, 521)$ | $0$ | $4$ |
Integral points
\( \left(-19, 9\right) \), \( \left(45, 521\right) \), \( \left(45, -567\right) \)
Invariants
| Conductor: | $N$ | = | \( 3570 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $-3563732336640$ | = | $-1 \cdot 2^{24} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 17^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{3168685387909439}{3563732336640} \) | = | $2^{-24} \cdot 3^{-1} \cdot 5^{-1} \cdot 7^{-2} \cdot 17^{-2} \cdot 191^{3} \cdot 769^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.0949381462501326578782011150$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0949381462501326578782011150$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9594119735978286$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.363165423342183$ | |||
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: | $\Omega$ | ≈ | $0.52560731487191860236313445267$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 96 $ = $ ( 2^{3} \cdot 3 )\cdot1\cdot1\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.1536438892315116141788067160 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 3.153643889 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.525607 \cdot 1.000000 \cdot 96}{4^2} \\ & \approx 3.153643889\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 9216 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $24$ | $I_{24}$ | split multiplicative | -1 | 1 | 24 | 24 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $7$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $17$ | $2$ | $I_{2}$ | 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 | 16.48.0.32 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 4065 & 16 \\ 4064 & 17 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 4076 & 4077 \end{array}\right),\left(\begin{array}{rr} 832 & 5 \\ 4035 & 4066 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 3060 & 3061 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 3982 & 4067 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 2561 & 16 \\ 3770 & 1311 \end{array}\right),\left(\begin{array}{rr} 1376 & 5 \\ 4035 & 4066 \end{array}\right),\left(\begin{array}{rr} 3133 & 16 \\ 3584 & 3765 \end{array}\right)$.
The torsion field $K:=\Q(E[4080])$ is a degree-$231022264320$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4080\Z)$.
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 | $4$ | \( 15 = 3 \cdot 5 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 595 = 5 \cdot 7 \cdot 17 \) |
| $5$ | split multiplicative | $6$ | \( 714 = 2 \cdot 3 \cdot 7 \cdot 17 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 3570t
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
This elliptic curve is its own minimal quadratic twist.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{17}) \) | \(\Z/8\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-255}) \) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-15}, \sqrt{17})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.7001316000000.25 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/16\Z\) | not in database |
| $8$ | deg 8 | \(\Z/16\Z\) | not in database |
| $8$ | 8.2.22202489064616875.6 | \(\Z/12\Z\) | 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 |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | 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 | 17 |
|---|---|---|---|---|---|
| Reduction type | split | nonsplit | split | nonsplit | split |
| $\lambda$-invariant(s) | 6 | 2 | 3 | 0 | 1 |
| $\mu$-invariant(s) | 0 | 0 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.