Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-4154700x-3259514000\)
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(homogenize, simplify) |
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\(y^2z=x^3-4154700xz^2-3259514000z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-4154700x-3259514000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-1180, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-1180:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-1180, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-1180, 0\right) \)
\([-1180:0:1]\)
\( \left(-1180, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 14400 \) | = | $2^{6} \cdot 3^{2} \cdot 5^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $100776960000000000$ | = | $2^{19} \cdot 3^{9} \cdot 5^{10} $ |
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| j-invariant: | $j$ | = | \( \frac{2656166199049}{33750} \) | = | $2^{-1} \cdot 3^{-3} \cdot 5^{-4} \cdot 11^{3} \cdot 1259^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4086057122733198317154701704$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.014859840882296834591619703139$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0501740895315954$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.987775902032563$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
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: | $\Omega$ | ≈ | $0.10574624830045968948568389673$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.3838799456147100635418846954 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 3.383879946 \approx L(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.105746 \cdot 1.000000 \cdot 32}{2^2} \\ & \approx 3.383879946\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 294912 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{9}^{*}$ | additive | 1 | 6 | 19 | 1 |
| $3$ | $2$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
| $5$ | $4$ | $I_{4}^{*}$ | additive | 1 | 2 | 10 | 4 |
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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 8.12.0.14 | $12$ |
| $3$ | 3B | 3.4.0.1 | $4$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 120 = 2^{3} \cdot 3 \cdot 5 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 1 & 12 \\ 12 & 25 \end{array}\right),\left(\begin{array}{rr} 97 & 24 \\ 96 & 25 \end{array}\right),\left(\begin{array}{rr} 104 & 39 \\ 25 & 94 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 23 & 96 \\ 36 & 71 \end{array}\right),\left(\begin{array}{rr} 114 & 71 \\ 85 & 64 \end{array}\right),\left(\begin{array}{rr} 94 & 3 \\ 15 & 34 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 14 & 11 \end{array}\right)$.
The torsion field $K:=\Q(E[120])$ is a degree-$92160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/120\Z)$.
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 | $4$ | \( 225 = 3^{2} \cdot 5^{2} \) |
| $3$ | additive | $2$ | \( 1600 = 2^{6} \cdot 5^{2} \) |
| $5$ | additive | $18$ | \( 576 = 2^{6} \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 14400bo
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 30a5, its twist by $-120$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{6}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-10}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-30}) \) | \(\Z/6\Z\) | 2.0.120.1-30.1-f7 |
| $4$ | \(\Q(\sqrt{6}, \sqrt{-10})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-5}, \sqrt{6})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{-10})\) | \(\Z/12\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{-15})\) | \(\Z/12\Z\) | not in database |
| $6$ | 6.2.57600000.1 | \(\Z/6\Z\) | not in database |
| $8$ | 8.4.7644119040000.3 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.29859840000.6 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.212336640000.28 | \(\Z/8\Z\) | not in database |
| $8$ | \(\Q(\sqrt{2}, \sqrt{3}, \sqrt{-5})\) | \(\Z/2\Z \oplus \Z/12\Z\) | 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 |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | 16.0.891610044825600000000.6 | \(\Z/2\Z \oplus \Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $18$ | 18.0.14572743731591310213120000000000000.1 | \(\Z/18\Z\) | 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 |
|---|---|---|---|
| Reduction type | add | add | add |
| $\lambda$-invariant(s) | - | - | - |
| $\mu$-invariant(s) | - | - | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ 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 $0$.