Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-1535x+23591\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-1535xz^2+23591z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-24555x+1485286\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{6}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(27, 22\right) \) | $0$ | $6$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([27:22:1]\) | $0$ | $6$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(107, 288\right) \) | $0$ | $6$ |
Integral points
\( \left(-45, 22\right) \), \( \left(19, 22\right) \), \( \left(19, -42\right) \), \( \left(27, 22\right) \), \( \left(27, -50\right) \)
\([-45:22:1]\), \([19:22:1]\), \([19:-42:1]\), \([27:22:1]\), \([27:-50:1]\)
\( \left(-181, 0\right) \), \((75,\pm 256)\), \((107,\pm 288)\)
Invariants
| Conductor: | $N$ | = | \( 126 \) | = | $2 \cdot 3^{2} \cdot 7$ |
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| Minimal Discriminant: | $\Delta$ | = | $-1337720832$ | = | $-1 \cdot 2^{18} \cdot 3^{6} \cdot 7 $ |
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| j-invariant: | $j$ | = | \( -\frac{548347731625}{1835008} \) | = | $-1 \cdot 2^{-18} \cdot 5^{3} \cdot 7^{-1} \cdot 1637^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.61583353447716664856978891257$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.066527390143111802872166294109$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0293250379726038$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.953195828202176$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $1.5305454480783489375703797074$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 36 $ = $ ( 2 \cdot 3^{2} )\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $6$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.5305454480783489375703797074 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.530545448 \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 1.530545 \cdot 1.000000 \cdot 36}{6^2} \\ & \approx 1.530545448\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 72 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $18$ | $I_{18}$ | split multiplicative | -1 | 1 | 18 | 18 |
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $7$ | $1$ | $I_{1}$ | split 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 | 8.6.0.1 | $6$ |
| $3$ | 3B.1.1 | 9.24.0.1 | $24$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \), index $864$, genus $21$, and generators
$\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 253 & 36 \\ 10 & 361 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 127 & 36 \\ 262 & 361 \end{array}\right),\left(\begin{array}{rr} 469 & 36 \\ 468 & 37 \end{array}\right),\left(\begin{array}{rr} 19 & 36 \\ 216 & 91 \end{array}\right),\left(\begin{array}{rr} 292 & 9 \\ 223 & 178 \end{array}\right),\left(\begin{array}{rr} 107 & 468 \\ 416 & 475 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 14 & 253 \end{array}\right)$.
The torsion field $K:=\Q(E[504])$ is a degree-$13934592$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/504\Z)$.
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 | $4$ | \( 63 = 3^{2} \cdot 7 \) |
| $3$ | additive | $6$ | \( 7 \) |
| $7$ | split multiplicative | $8$ | \( 18 = 2 \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 6, 9 and 18.
Its isogeny class 126.b
consists of 6 curves linked by isogenies of
degrees dividing 18.
Twists
The minimal quadratic twist of this elliptic curve is 14.a2, its twist by $-3$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | 2.0.7.1-2268.2-b1 |
| $3$ | 3.3.3969.1 | \(\Z/18\Z\) | not in database |
| $4$ | 4.2.4032.1 | \(\Z/12\Z\) | not in database |
| $6$ | 6.0.5250987.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.964467.3 | \(\Z/18\Z\) | not in database |
| $6$ | 6.0.110270727.2 | \(\Z/2\Z \oplus \Z/18\Z\) | not in database |
| $8$ | 8.0.9758278656.4 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | 8.0.796594176.12 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $12$ | 12.0.1351070359234281.1 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $12$ | 12.0.2233402022407689.1 | \(\Z/2\Z \oplus \Z/18\Z\) | not in database |
| $12$ | deg 12 | \(\Z/36\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $18$ | 18.0.8549394874383196572862347.3 | \(\Z/3\Z \oplus \Z/18\Z\) | 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 | 7 |
|---|---|---|---|
| Reduction type | split | add | split |
| $\lambda$-invariant(s) | 2 | - | 1 |
| $\mu$-invariant(s) | 0 | - | 0 |
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 $0$.