Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-60768x+5741812\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-60768xz^2+5741812z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-4922235x+4200547626\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(146, 56)$ | $1.4395665893312707763023864132$ | $\infty$ |
| $(139, 0)$ | $0$ | $2$ |
Integral points
\((-222,\pm 2888)\), \( \left(139, 0\right) \), \((146,\pm 56)\), \((668,\pm 16238)\)
Invariants
| Conductor: | $N$ | = | \( 40432 \) | = | $2^{4} \cdot 7 \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $18884593000448$ | = | $2^{13} \cdot 7^{2} \cdot 19^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{128787625}{98} \) | = | $2^{-1} \cdot 5^{3} \cdot 7^{-2} \cdot 101^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.4798553618981403056072829553$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.68551130824502523381446288210$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9676277689392396$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.2100960536074705$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.4395665893312707763023864132$ |
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| Real period: | $\Omega$ | ≈ | $0.68182652834942754949808048840$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2^{2}\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.9261387597262656835338253302 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.926138760 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.681827 \cdot 1.439567 \cdot 16}{2^2} \\ & \approx 3.926138760\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 114048 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{5}^{*}$ | additive | -1 | 4 | 13 | 1 |
| $7$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $19$ | $2$ | $I_0^{*}$ | 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 | 8.6.0.6 |
| $3$ | 3B | 9.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 \), index $864$, genus $21$, and generators
$\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 2393 & 7524 \\ 5282 & 8207 \end{array}\right),\left(\begin{array}{rr} 1825 & 2052 \\ 1938 & 2281 \end{array}\right),\left(\begin{array}{rr} 9541 & 36 \\ 9540 & 37 \end{array}\right),\left(\begin{array}{rr} 19 & 36 \\ 3240 & 6139 \end{array}\right),\left(\begin{array}{rr} 1540 & 1539 \\ 57 & 7354 \end{array}\right),\left(\begin{array}{rr} 8567 & 0 \\ 0 & 9575 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 14 & 253 \end{array}\right),\left(\begin{array}{rr} 5321 & 2052 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[9576])$ is a degree-$1715626967040$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9576\Z)$.
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 | $4$ | \( 361 = 19^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 5776 = 2^{4} \cdot 19^{2} \) |
| $19$ | additive | $182$ | \( 112 = 2^{4} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 6, 9 and 18.
Its isogeny class 40432r
consists of 6 curves linked by isogenies of
degrees dividing 18.
Twists
The minimal quadratic twist of this elliptic curve is 14a6, its twist by $76$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{19}) \) | \(\Z/6\Z\) | 2.2.76.1-98.1-a5 |
| $4$ | 4.0.141512.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{19})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.113829988608.10 | \(\Z/6\Z\) | not in database |
| $6$ | 6.6.1053981376.1 | \(\Z/18\Z\) | not in database |
| $8$ | 8.4.107134754750464.5 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1281641353216.14 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1281641353216.13 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | 12.0.5396612372552024064.1 | \(\Z/18\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | 12.12.4550151130951079428096.1 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 |
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 | ord | ss | nonsplit | ss | ord | ord | add | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 3 | 1,1 | 1 | 3,1 | 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 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.