Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-999296x+377303040\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-999296xz^2+377303040z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1295087643x+17607335897142\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{6}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(640, -320)$ | $0$ | $2$ |
| $(388, 6736)$ | $0$ | $6$ |
Integral points
\( \left(-1152, 576\right) \), \( \left(-272, 25216\right) \), \( \left(-272, -24944\right) \), \( \left(388, 6736\right) \), \( \left(388, -7124\right) \), \( \left(640, -320\right) \), \( \left(784, 8320\right) \), \( \left(784, -9104\right) \), \( \left(2236, 95440\right) \), \( \left(2236, -97676\right) \)
Invariants
| Conductor: | $N$ | = | \( 43890 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $2339305154932838400$ | = | $2^{12} \cdot 3^{6} \cdot 5^{2} \cdot 7^{2} \cdot 11^{6} \cdot 19^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{110358600993178429667329}{2339305154932838400} \) | = | $2^{-12} \cdot 3^{-6} \cdot 5^{-2} \cdot 7^{-2} \cdot 11^{-6} \cdot 19^{-2} \cdot 97^{3} \cdot 494497^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3140236103968542144667823576$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.3140236103968542144667823576$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9666984951419172$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.963591467549364$ | |||
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: | $\Omega$ | ≈ | $0.25857628357922603958578293419$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 3456 $ = $ ( 2^{2} \cdot 3 )\cdot( 2 \cdot 3 )\cdot2\cdot2\cdot( 2 \cdot 3 )\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $12$ |
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| Special value: | $ L(E,1)$ | ≈ | $6.2058308059014249500587904205 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 6.205830806 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.258576 \cdot 1.000000 \cdot 3456}{12^2} \\ & \approx 6.205830806\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 995328 |
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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 semistable. There are 6 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
| $3$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $7$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $11$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $19$ | $2$ | $I_{2}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2Cs | 2.6.0.1 | $6$ |
| $3$ | 3B.1.1 | 3.8.0.1 | $8$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 87780 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 19 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 17557 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 9 & 4 \\ 87764 & 87773 \end{array}\right),\left(\begin{array}{rr} 43897 & 12 \\ 87714 & 87667 \end{array}\right),\left(\begin{array}{rr} 87769 & 12 \\ 87768 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 58527 & 10 \\ 58522 & 3 \end{array}\right),\left(\begin{array}{rr} 37621 & 12 \\ 50166 & 73 \end{array}\right),\left(\begin{array}{rr} 71821 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 69307 & 6 \\ 78534 & 87775 \end{array}\right)$.
The torsion field $K:=\Q(E[87780])$ is a degree-$18871896637440000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/87780\Z)$.
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 | $4$ | \( 1 \) |
| $3$ | split multiplicative | $4$ | \( 665 = 5 \cdot 7 \cdot 19 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 8778 = 2 \cdot 3 \cdot 7 \cdot 11 \cdot 19 \) |
| $7$ | split multiplicative | $8$ | \( 6270 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 19 \) |
| $11$ | split multiplicative | $12$ | \( 3990 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 19 \) |
| $19$ | split multiplicative | $20$ | \( 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 43890.ct
consists of 8 curves linked by isogenies of
degrees dividing 12.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{6}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{55}, \sqrt{-57})\) | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $4$ | \(\Q(\sqrt{7}, \sqrt{57})\) | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-7}, \sqrt{-55})\) | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $6$ | 6.0.5280199666875.2 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $9$ | 9.3.318157002959141467570680000.1 | \(\Z/2\Z \oplus \Z/18\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | 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 | 11 | 19 |
|---|---|---|---|---|---|---|
| Reduction type | split | split | nonsplit | split | split | split |
| $\lambda$-invariant(s) | 4 | 3 | 0 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0 | 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 $0$.