Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-5x+2\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-5xz^2+2z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-6507x+199206\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-3, 1)$ | $0$ | $2$ |
| $(2, 1)$ | $0$ | $4$ |
Integral points
\( \left(-3, 1\right) \), \( \left(0, 1\right) \), \( \left(0, -2\right) \), \( \left(1, -1\right) \), \( \left(2, 1\right) \), \( \left(2, -4\right) \)
Invariants
| Conductor: | $N$ | = | \( 15 \) | = | $3 \cdot 5$ |
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| Discriminant: | $\Delta$ | = | $225$ | = | $3^{2} \cdot 5^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{13997521}{225} \) | = | $3^{-2} \cdot 5^{-2} \cdot 241^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.74885116236281644215221255789$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.74885116236281644215221255789$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9622954410890624$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.076102575120219$ | |||
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: | $\Omega$ | ≈ | $5.6024121693304080927207233472$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $8$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.35015076058315050579504520920 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.350150761 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 5.602412 \cdot 1.000000 \cdot 4}{8^2} \\ & \approx 0.350150761\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 2 |
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Local data at primes of bad reduction
This elliptic curve is semistable. There are 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $5$ | $2$ | $I_{2}$ | 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 |
|---|---|---|
| $2$ | 2Cs | 16.96.0.7 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 240 = 2^{4} \cdot 3 \cdot 5 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 161 & 16 \\ 6 & 97 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 16 \\ 4 & 193 \end{array}\right),\left(\begin{array}{rr} 61 & 8 \\ 170 & 197 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 4 & 65 \end{array}\right),\left(\begin{array}{rr} 225 & 16 \\ 224 & 17 \end{array}\right),\left(\begin{array}{rr} 15 & 8 \\ 208 & 209 \end{array}\right)$.
The torsion field $K:=\Q(E[240])$ is a degree-$737280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/240\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 1 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 5 \) |
| $5$ | split multiplicative | $6$ | \( 3 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 15a
consists of 8 curves linked by isogenies of
degrees dividing 16.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{5}) \) | \(\Z/2\Z \oplus \Z/8\Z\) | 2.2.5.1-45.1-a6 |
| $4$ | \(\Q(i, \sqrt{15})\) | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\zeta_{12})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $4$ | \(\Q(\zeta_{15})^+\) | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $8$ | 8.0.12960000.1 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $8$ | \(\Q(\zeta_{20})\) | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $8$ | 8.2.110716875.2 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | 16.0.1426576071720960000.2 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | \(\Q(\zeta_{60})\) | \(\Z/4\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/32\Z\) | 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 |
|---|---|---|---|
| Reduction type | ord | nonsplit | split |
| $\lambda$-invariant(s) | 0 | 0 | 1 |
| $\mu$-invariant(s) | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.