Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-460x-3830\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-460xz^2-3830z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-595539x-176894226\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 26 \) | = | $2 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-6656$ | = | $-1 \cdot 2^{9} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( -\frac{10730978619193}{6656} \) | = | $-1 \cdot 2^{-9} \cdot 7^{3} \cdot 13^{-1} \cdot 23^{3} \cdot 137^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.054386426133376655773210510687$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.054386426133376655773210510687$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0219339430875438$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $9.209106174142823$ | |||
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: | $\Omega$ | ≈ | $0.51557665127729445087676840909$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.51557665127729445087676840909 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.515576651 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.515577 \cdot 1.000000 \cdot 1}{1^2} \\ & \approx 0.515576651\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6 |
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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 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{9}$ | nonsplit multiplicative | 1 | 1 | 9 | 9 |
| $13$ | $1$ | $I_{1}$ | 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 |
|---|---|---|---|
| $3$ | 3B.1.2 | 9.24.0.3 | $24$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 478 & 9 \\ 225 & 928 \end{array}\right),\left(\begin{array}{rr} 448 & 9 \\ 855 & 76 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 459 & 928 \end{array}\right),\left(\begin{array}{rr} 919 & 18 \\ 918 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 235 & 486 \\ 0 & 443 \end{array}\right)$.
The torsion field $K:=\Q(E[936])$ is a degree-$1086898176$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/936\Z)$.
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 | $4$ | \( 13 \) |
| $3$ | good | $2$ | \( 13 \) |
| $13$ | split multiplicative | $14$ | \( 2 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 26a
consists of 3 curves linked by isogenies of
degrees dividing 9.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z\) | 2.0.3.1-676.2-b1 |
| $3$ | 3.1.104.1 | \(\Z/2\Z\) | not in database |
| $3$ | 3.1.4563.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.1124864.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.62462907.1 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $6$ | 6.0.562166163.2 | \(\Z/9\Z\) | not in database |
| $6$ | 6.0.3326427.2 | \(\Z/9\Z\) | not in database |
| $6$ | 6.0.292032.1 | \(\Z/6\Z\) | not in database |
| $9$ | 9.1.632360478776832.1 | \(\Z/6\Z\) | not in database |
| $12$ | 12.2.8421963387109376.8 | \(\Z/4\Z\) | not in database |
| $12$ | 12.0.922417564483584.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.14390607364515336591749112507.3 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
| $18$ | 18.0.10796753928209333245961218818048.2 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.7870833613664603936305728518356992.1 | \(\Z/18\Z\) | not in database |
| $18$ | 18.0.275579763091789640989661724672.1 | \(\Z/18\Z\) | not in database |
| $18$ | 18.0.2661599783191160077226587868626944.2 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
We only show fields where the torsion growth is primitive.
Iwasawa invariants
| $p$ | 2 | 3 | 13 |
|---|---|---|---|
| Reduction type | nonsplit | ord | split |
| $\lambda$-invariant(s) | 1 | 0 | 1 |
| $\mu$-invariant(s) | 0 | 2 | 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 $0$.