Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-140712033x+553560263937\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-140712033xz^2+553560263937z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-11397674700x+403511239386000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(4647, 0)$ | $0$ | $2$ |
Integral points
\( \left(4647, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 91200 \) | = | $2^{6} \cdot 3 \cdot 5^{2} \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $45954293760000000000000000$ | = | $2^{28} \cdot 3^{10} \cdot 5^{16} \cdot 19 $ |
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| j-invariant: | $j$ | = | \( \frac{75224183150104868881}{11219310000000000} \) | = | $2^{-10} \cdot 3^{-10} \cdot 5^{-10} \cdot 19^{-1} \cdot 4221361^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.6485498918519559459361018778$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.8041101647949877945098740290$ |
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| $abc$ quality: | $Q$ | ≈ | $1.032281228728047$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.945312120202671$ | |||
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: | $\Omega$ | ≈ | $0.061212051981364695180210081507$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot2\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.9587856634036702457667226082 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 1.958785663 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.061212 \cdot 1.000000 \cdot 32}{2^2} \\ & \approx 1.958785663\end{aligned}$$
Modular invariants
Modular form 91200.2.a.bn
For more coefficients, see the Downloads section to the right.
| Modular degree: | 22118400 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{18}^{*}$ | additive | -1 | 6 | 28 | 10 |
| $3$ | $2$ | $I_{10}$ | nonsplit multiplicative | 1 | 1 | 10 | 10 |
| $5$ | $4$ | $I_{10}^{*}$ | additive | 1 | 2 | 16 | 10 |
| $19$ | $1$ | $I_{1}$ | 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 | 2.3.0.1 |
| $5$ | 5B.4.1 | 5.12.0.1 |
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 $288$, genus $5$, and generators
$\left(\begin{array}{rr} 2049 & 2260 \\ 2090 & 2153 \end{array}\right),\left(\begin{array}{rr} 2261 & 20 \\ 2260 & 21 \end{array}\right),\left(\begin{array}{rr} 1139 & 0 \\ 0 & 2279 \end{array}\right),\left(\begin{array}{rr} 2269 & 2264 \\ 1380 & 1489 \end{array}\right),\left(\begin{array}{rr} 761 & 20 \\ 770 & 201 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 2040 & 1931 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right),\left(\begin{array}{rr} 2274 & 2275 \\ 1655 & 2234 \end{array}\right),\left(\begin{array}{rr} 1096 & 5 \\ 195 & 1126 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right),\left(\begin{array}{rr} 1159 & 2260 \\ 1160 & 2259 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[2280])$ is a degree-$15128985600$ 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 | $2$ | \( 475 = 5^{2} \cdot 19 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 30400 = 2^{6} \cdot 5^{2} \cdot 19 \) |
| $5$ | additive | $14$ | \( 1216 = 2^{6} \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 5 and 10.
Its isogeny class 91200.bn
consists of 4 curves linked by isogenies of
degrees dividing 10.
Twists
The minimal quadratic twist of this elliptic curve is 570.l3, its twist by $-40$.
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 |
| $2$ | \(\Q(\sqrt{-10}) \) | \(\Z/10\Z\) | not in database |
| $4$ | 4.0.273600.4 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-10}, \sqrt{19})\) | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.432373800960000.14 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.299427840000.27 | \(\Z/20\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/20\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/20\Z\) | not in database |
| $16$ | deg 16 | \(\Z/30\Z\) | not in database |
| $20$ | 20.4.9451648239783810422571008000000000000000.1 | \(\Z/10\Z\) | 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 | nonsplit | add | nonsplit |
| $\lambda$-invariant(s) | - | 0 | - | 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 $0$.