Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-3268x-71074\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-3268xz^2-71074z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-52283x-4601002\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-33, 16)$ | $0$ | $2$ |
Integral points
\( \left(-33, 16\right) \)
Invariants
| Conductor: | $N$ | = | \( 3185 \) | = | $5 \cdot 7^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $374712065$ | = | $5 \cdot 7^{8} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( \frac{32798729601}{3185} \) | = | $3^{3} \cdot 5^{-1} \cdot 7^{-2} \cdot 11^{3} \cdot 13^{-1} \cdot 97^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.68093657587617891042410490556$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.29201849865147774212857146616$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8827733875533473$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.449317254937965$ | |||
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: | $\Omega$ | ≈ | $0.63145434511237203732110083119$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 1\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.63145434511237203732110083119 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.631454345 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.631454 \cdot 1.000000 \cdot 4}{2^2} \\ & \approx 0.631454345\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1536 |
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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 not semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $7$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $13$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
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 | 16.24.0.18 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 7280 = 2^{4} \cdot 5 \cdot 7 \cdot 13 \), index $192$, genus $3$, and generators
$\left(\begin{array}{rr} 3119 & 7264 \\ 3112 & 7151 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 3656 \\ 4550 & 911 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 8 & 65 \end{array}\right),\left(\begin{array}{rr} 7265 & 16 \\ 7264 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 5460 & 5461 \end{array}\right),\left(\begin{array}{rr} 574 & 3 \\ 1213 & 20 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 5838 & 3 \\ 4461 & 20 \end{array}\right),\left(\begin{array}{rr} 15 & 166 \\ 6994 & 4115 \end{array}\right)$.
The torsion field $K:=\Q(E[7280])$ is a degree-$3246202552320$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/7280\Z)$.
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 |
|---|---|---|---|
| $5$ | nonsplit multiplicative | $6$ | \( 637 = 7^{2} \cdot 13 \) |
| $7$ | additive | $32$ | \( 65 = 5 \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 245 = 5 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 3185a
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 455b1, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{65}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/4\Z\) | 2.0.7.1-29575.1-a4 |
| $2$ | \(\Q(\sqrt{-455}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-7}, \sqrt{65})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.46356673636000000.26 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.2596921600.3 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.181080756390625.2 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | 5 | 7 | 13 |
|---|---|---|---|---|
| Reduction type | ord | nonsplit | add | nonsplit |
| $\lambda$-invariant(s) | 4 | 0 | - | 0 |
| $\mu$-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 $0$.