Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-28x+272\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-28xz^2+272z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-36315x+12799350\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{10}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(2, 14\right) \) | $0$ | $10$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([2:14:1]\) | $0$ | $10$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(75, 3240\right) \) | $0$ | $10$ |
Integral points
\( \left(-8, 4\right) \), \( \left(-4, 20\right) \), \( \left(-4, -16\right) \), \( \left(2, 14\right) \), \( \left(2, -16\right) \), \( \left(8, 20\right) \), \( \left(8, -28\right) \), \( \left(32, 164\right) \), \( \left(32, -196\right) \)
\([-8:4:1]\), \([-4:20:1]\), \([-4:-16:1]\), \([2:14:1]\), \([2:-16:1]\), \([8:20:1]\), \([8:-28:1]\), \([32:164:1]\), \([32:-196:1]\)
\( \left(-285, 0\right) \), \((-141,\pm 3888)\), \((75,\pm 3240)\), \((291,\pm 5184)\), \((1155,\pm 38880)\)
Invariants
| Conductor: | $N$ | = | \( 150 \) | = | $2 \cdot 3 \cdot 5^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-31104000$ | = | $-1 \cdot 2^{10} \cdot 3^{5} \cdot 5^{3} $ |
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| j-invariant: | $j$ | = | \( -\frac{19465109}{248832} \) | = | $-1 \cdot 2^{-10} \cdot 3^{-5} \cdot 269^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.11944777360291933688272067076$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.28291170450560575676746916255$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0975364253149262$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.939860985040359$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
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: | $\Omega$ | ≈ | $1.7692592482524138311369605540$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 100 $ = $ ( 2 \cdot 5 )\cdot5\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $10$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.7692592482524138311369605540 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.769259248 \approx L(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.769259 \cdot 1.000000 \cdot 100}{10^2} \\ & \approx 1.769259248\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 40 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $10$ | $I_{10}$ | split multiplicative | -1 | 1 | 10 | 10 |
| $3$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $5$ | $2$ | $III$ | additive | -1 | 2 | 3 | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
| $5$ | 5B.1.1 | 5.24.0.1 | $24$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 120 = 2^{3} \cdot 3 \cdot 5 \), index $288$, genus $5$, and generators
$\left(\begin{array}{rr} 61 & 20 \\ 10 & 81 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 0 & 11 \end{array}\right),\left(\begin{array}{rr} 101 & 20 \\ 100 & 21 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 31 & 20 \\ 70 & 81 \end{array}\right),\left(\begin{array}{rr} 56 & 5 \\ 75 & 106 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right),\left(\begin{array}{rr} 66 & 5 \\ 55 & 82 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right)$.
The torsion field $K:=\Q(E[120])$ is a degree-$122880$ 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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 15 = 3 \cdot 5 \) |
| $3$ | split multiplicative | $4$ | \( 50 = 2 \cdot 5^{2} \) |
| $5$ | additive | $10$ | \( 1 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 5 and 10.
Its isogeny class 150a
consists of 4 curves linked by isogenies of
degrees dividing 10.
Twists
This elliptic curve is its own minimal quadratic twist.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{10}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/2\Z \oplus \Z/10\Z\) | 2.0.15.1-300.2-b4 |
| $4$ | \(\Q(\sqrt{-5 +2 \sqrt{10}})\) | \(\Z/20\Z\) | not in database |
| $8$ | 8.0.11664000000.8 | \(\Z/2\Z \oplus \Z/20\Z\) | not in database |
| $8$ | 8.0.5184000000.10 | \(\Z/2\Z \oplus \Z/20\Z\) | not in database |
| $8$ | 8.2.44286750000.2 | \(\Z/30\Z\) | not in database |
| $16$ | deg 16 | \(\Z/40\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/30\Z\) | not in database |
| $20$ | 20.0.4656612873077392578125.1 | \(\Z/5\Z \oplus \Z/10\Z\) | 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 |
|---|---|---|---|
| Reduction type | split | split | add |
| $\lambda$-invariant(s) | 1 | 1 | - |
| $\mu$-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 $0$.