Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-5709499521x-166050503697375\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-5709499521xz^2-166050503697375z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-462469461228x-121052204603770032\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-43625, 0)$ | $0$ | $2$ |
Integral points
\( \left(-43625, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 73920 \) | = | $2^{6} \cdot 3 \cdot 5 \cdot 7 \cdot 11$ |
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| Discriminant: | $\Delta$ | = | $25278036552984821760$ | = | $2^{28} \cdot 3^{3} \cdot 5 \cdot 7^{8} \cdot 11^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{78519570041710065450485106721}{96428056919040} \) | = | $2^{-10} \cdot 3^{-3} \cdot 5^{-1} \cdot 7^{-8} \cdot 11^{-2} \cdot 2543^{3} \cdot 1683887^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.8971527625921347738220960757$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.8574319917522168096962478935$ |
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| $abc$ quality: | $Q$ | ≈ | $1.052918061430929$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.047688535459382$ | |||
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: | $\Omega$ | ≈ | $0.017368009358423927680970915708$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot1\cdot1\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.13894407486739142144776732567 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 0.138944075 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.017368 \cdot 1.000000 \cdot 8}{2^2} \\ & \approx 0.138944075\end{aligned}$$
Modular invariants
Modular form 73920.2.a.n
For more coefficients, see the Downloads section to the right.
| Modular degree: | 35389440 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{18}^{*}$ | additive | -1 | 6 | 28 | 10 |
| $3$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $7$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $11$ | $2$ | $I_{2}$ | nonsplit 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 | 8.48.0.225 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1016 & 1 \\ 1423 & 10 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 1676 & 1677 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1665 & 16 \\ 1664 & 17 \end{array}\right),\left(\begin{array}{rr} 1462 & 409 \\ 293 & 954 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 1582 & 1667 \end{array}\right),\left(\begin{array}{rr} 241 & 16 \\ 248 & 129 \end{array}\right),\left(\begin{array}{rr} 568 & 1 \\ 1199 & 10 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 1667 & 1664 \\ 996 & 935 \end{array}\right)$.
The torsion field $K:=\Q(E[1680])$ is a degree-$5945425920$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1680\Z)$.
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 | $2$ | \( 15 = 3 \cdot 5 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 24640 = 2^{6} \cdot 5 \cdot 7 \cdot 11 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 14784 = 2^{6} \cdot 3 \cdot 7 \cdot 11 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 10560 = 2^{6} \cdot 3 \cdot 5 \cdot 11 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 6720 = 2^{6} \cdot 3 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 73920.n
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 2310.o1, its twist by $-8$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{15}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-30}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{15})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\zeta_{8})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-15})\) | \(\Z/8\Z\) | not in database |
| $8$ | 8.4.10929447936000000.41 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.2732361984000000.28 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.3317760000.4 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | 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 |
|---|---|---|---|---|---|
| Reduction type | add | nonsplit | nonsplit | nonsplit | nonsplit |
| $\lambda$-invariant(s) | - | 0 | 0 | 0 | 0 |
| $\mu$-invariant(s) | - | 0 | 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 $0$.