Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-285939008x-2079965759488\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-285939008xz^2-2079965759488z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-23161059675x-1516364521845750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(19777, 0)$ | $0$ | $2$ |
Integral points
\( \left(19777, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 115600 \) | = | $2^{4} \cdot 5^{2} \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $-372878231827047040000000000$ | = | $-1 \cdot 2^{16} \cdot 5^{10} \cdot 17^{12} $ |
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| j-invariant: | $j$ | = | \( -\frac{1673672305534489}{241375690000} \) | = | $-1 \cdot 2^{-4} \cdot 5^{-4} \cdot 13^{3} \cdot 17^{-6} \cdot 9133^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.8292167317930545412184327561$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.91474392298795100437605365909$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9921013018955642$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.025959657896066$ | |||
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: | $\Omega$ | ≈ | $0.018211498990347683386406042131$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2^{2}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.6422997980695366772812084263 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $25$ = $5^2$ (exact) |
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BSD formula
$$\begin{aligned} 3.642299798 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{25 \cdot 0.018211 \cdot 1.000000 \cdot 32}{2^2} \\ & \approx 3.642299798\end{aligned}$$
Modular invariants
Modular form 115600.2.a.ct
For more coefficients, see the Downloads section to the right.
| Modular degree: | 53084160 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{8}^{*}$ | additive | -1 | 4 | 16 | 4 |
| $5$ | $4$ | $I_{4}^{*}$ | additive | 1 | 2 | 10 | 4 |
| $17$ | $4$ | $I_{6}^{*}$ | additive | 1 | 2 | 12 | 6 |
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 | 4.6.0.5 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2040 = 2^{3} \cdot 3 \cdot 5 \cdot 17 \), index $384$, genus $9$, and generators
$\left(\begin{array}{rr} 2017 & 24 \\ 2016 & 25 \end{array}\right),\left(\begin{array}{rr} 465 & 2018 \\ 658 & 1597 \end{array}\right),\left(\begin{array}{rr} 13 & 24 \\ 1332 & 733 \end{array}\right),\left(\begin{array}{rr} 1529 & 2016 \\ 1779 & 1895 \end{array}\right),\left(\begin{array}{rr} 1021 & 24 \\ 6 & 145 \end{array}\right),\left(\begin{array}{rr} 699 & 16 \\ 1846 & 123 \end{array}\right),\left(\begin{array}{rr} 15 & 16 \\ 314 & 335 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 807 & 2036 \\ 1348 & 55 \end{array}\right)$.
The torsion field $K:=\Q(E[2040])$ is a degree-$7219445760$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2040\Z)$.
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 | $2$ | \( 7225 = 5^{2} \cdot 17^{2} \) |
| $5$ | additive | $18$ | \( 4624 = 2^{4} \cdot 17^{2} \) |
| $17$ | additive | $162$ | \( 400 = 2^{4} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 115600by
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 170b2, its twist by $-340$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-85}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.2.28900.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{85})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.106120800000.2 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.13363360000.2 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.3421020160000.33 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.13363360000.3 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | 16.0.178579390489600000000.1 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.0.10683936791299357251056640000000000000.1 | \(\Z/18\Z\) | 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 | 17 |
|---|---|---|---|---|
| Reduction type | add | ord | add | add |
| $\lambda$-invariant(s) | - | 0 | - | - |
| $\mu$-invariant(s) | - | 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 $0$.