Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-16x+4096\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-16xz^2+4096z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1323x+2982042\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(0, 64)$ | $1.8482380023883666572805746492$ | $\infty$ |
| $(-16, 0)$ | $0$ | $2$ |
Integral points
\( \left(-16, 0\right) \), \((0,\pm 64)\), \((33,\pm 196)\)
Invariants
| Conductor: | $N$ | = | \( 11760 \) | = | $2^{4} \cdot 3 \cdot 5 \cdot 7^{2}$ |
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| Discriminant: | $\Delta$ | = | $-7228354560$ | = | $-1 \cdot 2^{12} \cdot 3 \cdot 5 \cdot 7^{6} $ |
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| j-invariant: | $j$ | = | \( -\frac{1}{15} \) | = | $-1 \cdot 3^{-1} \cdot 5^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.57067750244481286510907987456$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.0954247526427890968608286186$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1980768440515948$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.217515817192996$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.8482380023883666572805746492$ |
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| Real period: | $\Omega$ | ≈ | $1.0587563815807245938379537514$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2^{2}\cdot1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.9136675594173874036326074794 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.913667559 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.058756 \cdot 1.848238 \cdot 8}{2^2} \\ & \approx 3.913667559\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6144 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}^{*}$ | additive | -1 | 4 | 12 | 0 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
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 | 32.48.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 2752 & 2429 \\ 1547 & 162 \end{array}\right),\left(\begin{array}{rr} 694 & 483 \\ 2765 & 2668 \end{array}\right),\left(\begin{array}{rr} 23 & 18 \\ 798 & 1355 \end{array}\right),\left(\begin{array}{rr} 3359 & 448 \\ 2310 & 1889 \end{array}\right),\left(\begin{array}{rr} 183 & 1442 \\ 154 & 463 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3329 & 32 \\ 3328 & 33 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 1439 & 0 \\ 0 & 3359 \end{array}\right),\left(\begin{array}{rr} 5 & 28 \\ 68 & 381 \end{array}\right)$.
The torsion field $K:=\Q(E[3360])$ is a degree-$23781703680$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3360\Z)$.
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 | $2$ | \( 735 = 3 \cdot 5 \cdot 7^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \) |
| $7$ | additive | $26$ | \( 240 = 2^{4} \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4, 8 and 16.
Its isogeny class 11760.p
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 15.a7, its twist by $28$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-105}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{7}) \) | \(\Z/4\Z\) | 2.2.28.1-225.1-a2 |
| $4$ | \(\Q(\sqrt{5}, \sqrt{7})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{7})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{7}, \sqrt{-15})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.437582250000.11 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.7001316000000.54 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.31116960000.9 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.8.777924000000.3 | \(\Z/16\Z\) | not in database |
| $8$ | 8.0.9604000000.3 | \(\Z/16\Z\) | 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 | \(\Z/16\Z\) | not in database |
| $16$ | 16.0.605165749776000000000000.9 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/32\Z\) | 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 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | nonsplit | nonsplit | add | ord | ord | ord | ord | ss | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 3 | 1 | - | 1 | 1 | 1 | 1 | 1,1 | 1 | 1,1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | 0 | 0 | - | 0 | 0 | 0 | 0 | 0,0 | 0 | 0,0 | 0 | 0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.