Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-27408x+1495188\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-27408xz^2+1495188z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2220075x+1096652250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(42, 648)$ | $0.44811768362898181507347265555$ | $\infty$ |
| $(-12, 1350)$ | $0$ | $4$ |
Integral points
\((-166,\pm 1224)\), \((-102,\pm 1800)\), \((-12,\pm 1350)\), \((42,\pm 648)\), \((63,\pm 150)\), \( \left(123, 0\right) \), \((138,\pm 600)\), \((204,\pm 2106)\), \((348,\pm 5850)\), \((1338,\pm 48600)\)
Invariants
| Conductor: | $N$ | = | \( 1200 \) | = | $2^{4} \cdot 3 \cdot 5^{2}$ |
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| Discriminant: | $\Delta$ | = | $340122240000000$ | = | $2^{13} \cdot 3^{12} \cdot 5^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{35578826569}{5314410} \) | = | $2^{-1} \cdot 3^{-12} \cdot 5^{-1} \cdot 11^{3} \cdot 13^{3} \cdot 23^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.5127259776592923313092314912$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.014859840882296834591619703129$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0339287621454463$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.961770395294205$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.44811768362898181507347265555$ |
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| Real period: | $\Omega$ | ≈ | $0.51804870109955817783027952757$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 192 $ = $ 2^{2}\cdot( 2^{2} \cdot 3 )\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.7857614073248413034447420956 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.785761407 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.518049 \cdot 0.448118 \cdot 192}{4^2} \\ & \approx 2.785761407\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4608 |
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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 |
| $3$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
| $5$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 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 | 4.12.0.7 |
| $3$ | 3B | 3.4.0.1 |
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
$\left(\begin{array}{rr} 89 & 96 \\ 90 & 119 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 25 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 97 & 24 \\ 96 & 25 \end{array}\right),\left(\begin{array}{rr} 42 & 79 \\ 77 & 92 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 117 \\ 3 & 86 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 14 & 11 \end{array}\right),\left(\begin{array}{rr} 61 & 24 \\ 92 & 29 \end{array}\right)$.
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 | $4$ | \( 25 = 5^{2} \) |
| $3$ | split multiplicative | $4$ | \( 400 = 2^{4} \cdot 5^{2} \) |
| $5$ | additive | $18$ | \( 48 = 2^{4} \cdot 3 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 1200p
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 30a4, its twist by $-20$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{10}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-5}) \) | \(\Z/12\Z\) | not in database |
| $4$ | 4.0.9000.1 | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-5})\) | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $6$ | 6.2.21600000.1 | \(\Z/12\Z\) | not in database |
| $8$ | 8.0.262144000000.9 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.5184000000.12 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.8.849346560000.5 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.5184000000.25 | \(\Z/24\Z\) | not in database |
| $12$ | 12.0.466560000000000.4 | \(\Z/3\Z \oplus \Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | not in database |
| $18$ | 18.0.1054162596324602880000000000000.1 | \(\Z/36\Z\) | 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 | split | add | ord | ss | ord | ord | ord | ss | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 2 | - | 1 | 1,3 | 1 | 3 | 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.