Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-66x-216\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-66xz^2-216z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-5373x-141372\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(21, 90)$ | $2.7077321322842255603795325494$ | $\infty$ |
| $(-4, 0)$ | $0$ | $2$ |
| $(9, 0)$ | $0$ | $2$ |
Integral points
\( \left(-6, 0\right) \), \( \left(-4, 0\right) \), \( \left(9, 0\right) \), \((21,\pm 90)\)
Invariants
| Conductor: | $N$ | = | \( 6240 \) | = | $2^{5} \cdot 3 \cdot 5 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $2433600$ | = | $2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{504358336}{38025} \) | = | $2^{6} \cdot 3^{-2} \cdot 5^{-2} \cdot 13^{-2} \cdot 199^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.029371143300508013491975127215$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.37594473358048066820059118794$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8613315992780841$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.7690139778092497$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.7077321322842255603795325494$ |
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| Real period: | $\Omega$ | ≈ | $1.6808628335731643632398454780$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.5513263044283697000553108219 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.551326304 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.680863 \cdot 2.707732 \cdot 16}{4^2} \\ & \approx 4.551326304\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 768 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III$ | additive | -1 | 5 | 6 | 0 |
| $3$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $13$ | $2$ | $I_{2}$ | 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 |
|---|---|---|
| $2$ | 2Cs | 8.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 937 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 521 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1081 & 4 \\ 602 & 9 \end{array}\right),\left(\begin{array}{rr} 783 & 2 \\ 1558 & 1559 \end{array}\right),\left(\begin{array}{rr} 1173 & 2 \\ 778 & 1559 \end{array}\right),\left(\begin{array}{rr} 1557 & 4 \\ 1556 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[1560])$ is a degree-$19322634240$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1560\Z)$.
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 | $2$ | \( 1 \) |
| $3$ | split multiplicative | $4$ | \( 2080 = 2^{5} \cdot 5 \cdot 13 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 1248 = 2^{5} \cdot 3 \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 480 = 2^{5} \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 6240ba
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 6240a1, its twist by $-4$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-15})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{13}, \sqrt{15})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{-13})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | 16.0.8979181539709000089600000000.18 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | split | nonsplit | ss | ss | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 2 | 1 | 1,1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 3 | 1 | 3 | 1 | 1,1 |
| $\mu$-invariant(s) | - | 0 | 0 | 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.