Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-69x-194\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-69xz^2-194z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-88803x-8773218\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{6}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-4, 6\right) \) | $0$ | $6$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-4:6:1]\) | $0$ | $6$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-141, 972\right) \) | $0$ | $6$ |
Integral points
\( \left(-4, 6\right) \), \( \left(-4, -3\right) \), \( \left(14, 33\right) \), \( \left(14, -48\right) \)
\([-4:6:1]\), \([-4:-3:1]\), \([14:33:1]\), \([14:-48:1]\)
\((-141,\pm 972)\), \((507,\pm 8748)\)
Invariants
| Conductor: | $N$ | = | \( 30 \) | = | $2 \cdot 3 \cdot 5$ |
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| Minimal Discriminant: | $\Delta$ | = | $5314410$ | = | $2 \cdot 3^{12} \cdot 5 $ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.014859840882296834591619703102$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.014859840882296834591619703102$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0339287621454463$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.143077514276536$ | |||
| 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: | $\Omega$ | ≈ | $1.6759741296207482247241140670$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 1\cdot( 2^{2} \cdot 3 )\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $6$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.55865804320691607490803802232 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.558658043 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.675974 \cdot 1.000000 \cdot 12}{6^2} \\ & \approx 0.558658043\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 8 |
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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 semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 8.12.0.7 | $12$ |
| $3$ | 3B.1.1 | 3.8.0.1 | $8$ |
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
$\left(\begin{array}{rr} 1 & 12 \\ 12 & 25 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 97 & 24 \\ 96 & 25 \end{array}\right),\left(\begin{array}{rr} 42 & 79 \\ 77 & 92 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 31 & 24 \\ 30 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 14 & 11 \end{array}\right),\left(\begin{array}{rr} 61 & 24 \\ 92 & 29 \end{array}\right),\left(\begin{array}{rr} 112 & 3 \\ 117 & 34 \end{array}\right)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 5 \) |
| $3$ | split multiplicative | $4$ | \( 10 = 2 \cdot 5 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 6 = 2 \cdot 3 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 30a
consists of 8 curves linked by isogenies of
degrees dividing 12.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-5}) \) | \(\Z/12\Z\) | 2.0.20.1-90.2-a4 |
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/12\Z\) | 2.0.8.1-450.2-a4 |
| $2$ | \(\Q(\sqrt{10}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | 2.2.40.1-90.1-f6 |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-5})\) | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $6$ | 6.0.270000.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.4.65536000000.4 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | 8.0.5184000000.25 | \(\Z/24\Z\) | not in database |
| $8$ | 8.0.8493465600.20 | \(\Z/24\Z\) | not in database |
| $9$ | 9.3.143489070000.1 | \(\Z/18\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $12$ | 12.0.466560000000000.4 | \(\Z/3\Z \oplus \Z/12\Z\) | not in database |
| $12$ | 12.0.1194393600000000.1 | \(\Z/3\Z \oplus \Z/12\Z\) | 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 |
| $18$ | 18.6.539731249318196674560000000000000.1 | \(\Z/2\Z \oplus \Z/18\Z\) | not in database |
| $18$ | 18.0.1054162596324602880000000000000.1 | \(\Z/36\Z\) | not in database |
| $18$ | 18.0.172713999781822935859200000000.1 | \(\Z/36\Z\) | 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 |
|---|---|---|---|
| Reduction type | nonsplit | split | nonsplit |
| $\lambda$-invariant(s) | 0 | 1 | 0 |
| $\mu$-invariant(s) | 1 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.