Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-316100x-43167433\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-316100xz^2-43167433z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-409665627x-2007874761354\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-1901/4, 1897/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 1470 \) | = | $2 \cdot 3 \cdot 5 \cdot 7^{2}$ |
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| Discriminant: | $\Delta$ | = | $1221310198432836750$ | = | $2 \cdot 3 \cdot 5^{3} \cdot 7^{18} $ |
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| j-invariant: | $j$ | = | \( \frac{29689921233686449}{10380965400750} \) | = | $2^{-1} \cdot 3^{-1} \cdot 5^{-3} \cdot 7^{-12} \cdot 23^{3} \cdot 13463^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1719810875635814113786189724$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1990260130359247588259426007$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0394034516317685$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.801717355305041$ | |||
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: | $\Omega$ | ≈ | $0.20712173548899524585815008096$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 1\cdot1\cdot3\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.4854608258679429502978009715 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 2.485460826 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.207122 \cdot 1.000000 \cdot 12}{2^2} \\ & \approx 2.485460826\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 27648 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $7$ | $4$ | $I_{12}^{*}$ | additive | -1 | 2 | 18 | 12 |
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 | 8.12.0.6 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 599 & 816 \\ 468 & 551 \end{array}\right),\left(\begin{array}{rr} 42 & 199 \\ 317 & 812 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 352 & 3 \\ 717 & 754 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 374 & 11 \end{array}\right),\left(\begin{array}{rr} 646 & 21 \\ 345 & 466 \end{array}\right),\left(\begin{array}{rr} 576 & 719 \\ 305 & 546 \end{array}\right),\left(\begin{array}{rr} 817 & 24 \\ 816 & 25 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[840])$ is a degree-$185794560$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/840\Z)$.
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 | $4$ | \( 735 = 3 \cdot 5 \cdot 7^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 98 = 2 \cdot 7^{2} \) |
| $5$ | split multiplicative | $6$ | \( 294 = 2 \cdot 3 \cdot 7^{2} \) |
| $7$ | additive | $32$ | \( 30 = 2 \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 1470m
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 210a7, its twist by $-7$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{30}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-14}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-105}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{21}) \) | \(\Z/6\Z\) | 2.2.21.1-2100.1-bk4 |
| $4$ | \(\Q(\sqrt{-14}, \sqrt{30})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.0.1505280.5 | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{21}, \sqrt{30})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-6}, \sqrt{-14})\) | \(\Z/12\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-5}, \sqrt{21})\) | \(\Z/12\Z\) | not in database |
| $6$ | 6.0.324060912.5 | \(\Z/6\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.112021056000000.18 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.2039281090560000.128 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.7965941760000.5 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | 8.0.20392810905600.59 | \(\Z/24\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | 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 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | not in database |
| $18$ | 18.6.183277652619589776530736563812500000000.4 | \(\Z/18\Z\) | 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 | 5 | 7 |
|---|---|---|---|---|
| Reduction type | split | nonsplit | split | add |
| $\lambda$-invariant(s) | 2 | 2 | 1 | - |
| $\mu$-invariant(s) | 2 | 0 | 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 $0$.