Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-266821068x-1071201902992\)
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(homogenize, simplify) |
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\(y^2z=x^3-266821068xz^2-1071201902992z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-266821068x-1071201902992\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-4316, 0)$ | $0$ | $2$ |
Integral points
\( \left(-4316, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 54720 \) | = | $2^{6} \cdot 3^{2} \cdot 5 \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $720030384000000000000000000$ | = | $2^{22} \cdot 3^{8} \cdot 5^{18} \cdot 19^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{10993009831928446009969}{3767761230468750000} \) | = | $2^{-4} \cdot 3^{-2} \cdot 5^{-18} \cdot 19^{-3} \cdot 23^{3} \cdot 47^{3} \cdot 67^{3} \cdot 307^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.8553863362595932956628848657$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.2663594210856204858394140651$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0531544430296054$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.399630467051713$ | |||
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: | $\Omega$ | ≈ | $0.038395847824297820471994154129$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2^{2}\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.8395847824297820471994154129 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $25$ = $5^2$ (exact) |
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BSD formula
$$\begin{aligned} 3.839584782 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{25 \cdot 0.038396 \cdot 1.000000 \cdot 16}{2^2} \\ & \approx 3.839584782\end{aligned}$$
Modular invariants
Modular form 54720.2.a.cf
For more coefficients, see the Downloads section to the right.
| Modular degree: | 26542080 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{12}^{*}$ | additive | 1 | 6 | 22 | 4 |
| $3$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $5$ | $2$ | $I_{18}$ | nonsplit multiplicative | 1 | 1 | 18 | 18 |
| $19$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
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 | 2.3.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 2269 & 2278 \\ 1190 & 1149 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1614 & 1223 \\ 1615 & 1234 \end{array}\right),\left(\begin{array}{rr} 970 & 3 \\ 573 & 1132 \end{array}\right),\left(\begin{array}{rr} 457 & 12 \\ 462 & 73 \end{array}\right),\left(\begin{array}{rr} 1139 & 0 \\ 0 & 2279 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 2230 & 2271 \end{array}\right),\left(\begin{array}{rr} 1519 & 2268 \\ 1514 & 2207 \end{array}\right),\left(\begin{array}{rr} 2269 & 12 \\ 2268 & 13 \end{array}\right),\left(\begin{array}{rr} 1151 & 2268 \\ 1152 & 2267 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[2280])$ is a degree-$45386956800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2280\Z)$.
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 | $2$ | \( 171 = 3^{2} \cdot 19 \) |
| $3$ | additive | $8$ | \( 64 = 2^{6} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 10944 = 2^{6} \cdot 3^{2} \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 54720.cf
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 570.k1, its twist by $-24$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{19}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.273600.2 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{19})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.362797056.5 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.432373800960000.68 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.299427840000.16 | \(\Z/12\Z\) | not in database |
| $12$ | 12.0.131621703842267136.93 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.6.978612409468667852526368334500302159872.2 | \(\Z/18\Z\) | 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 | 19 |
|---|---|---|---|---|
| Reduction type | add | add | nonsplit | nonsplit |
| $\lambda$-invariant(s) | - | - | 2 | 0 |
| $\mu$-invariant(s) | - | - | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ 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 $0$.