Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-13x+22\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-13xz^2+22z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1080x+12825\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(1, 3\right) \) | $0.12118393281215308617449765095$ | $\infty$ |
| \( \left(2, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([1:3:1]\) | $0.12118393281215308617449765095$ | $\infty$ |
| \([2:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(6, 81\right) \) | $0.12118393281215308617449765095$ | $\infty$ |
| \( \left(15, 0\right) \) | $0$ | $2$ |
Integral points
\((-3,\pm 5)\), \((-2,\pm 6)\), \((1,\pm 3)\), \( \left(2, 0\right) \), \((3,\pm 1)\), \((7,\pm 15)\), \((11,\pm 33)\), \((127,\pm 1425)\)
\([-3:\pm 5:1]\), \([-2:\pm 6:1]\), \([1:\pm 3:1]\), \([2:0:1]\), \([3:\pm 1:1]\), \([7:\pm 15:1]\), \([11:\pm 33:1]\), \([127:\pm 1425:1]\)
\((-3,\pm 5)\), \((-2,\pm 6)\), \((1,\pm 3)\), \( \left(2, 0\right) \), \((3,\pm 1)\), \((7,\pm 15)\), \((11,\pm 33)\), \((127,\pm 1425)\)
Invariants
| Conductor: | $N$ | = | \( 300 \) | = | $2^{2} \cdot 3 \cdot 5^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $18000$ | = | $2^{4} \cdot 3^{2} \cdot 5^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{131072}{9} \) | = | $2^{17} \cdot 3^{-2}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.43525232363425922002740348808$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.0686608619294327501500040285$ |
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| $abc$ quality: | $Q$ | ≈ | $1.2015530913851162$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.398517495222249$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.12118393281215308617449765095$ |
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| Real period: | $\Omega$ | ≈ | $3.8069569608488996998681926817$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 3\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.3840260496868147045717027664 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.384026050 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 3.806957 \cdot 0.121184 \cdot 12}{2^2} \\ & \approx 1.384026050\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 24 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $5$ | $2$ | $III$ | 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 | 8.12.0.32 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 240 = 2^{4} \cdot 3 \cdot 5 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 181 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 225 & 16 \\ 224 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 8 \\ 112 & 147 \end{array}\right),\left(\begin{array}{rr} 11 & 12 \\ 140 & 131 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 13 & 8 \\ 8 & 5 \end{array}\right),\left(\begin{array}{rr} 161 & 10 \\ 94 & 141 \end{array}\right),\left(\begin{array}{rr} 151 & 10 \\ 134 & 141 \end{array}\right),\left(\begin{array}{rr} 204 & 7 \\ 233 & 52 \end{array}\right)$.
The torsion field $K:=\Q(E[240])$ is a degree-$5898240$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/240\Z)$.
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 | $2$ | \( 5 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 100 = 2^{2} \cdot 5^{2} \) |
| $5$ | additive | $10$ | \( 12 = 2^{2} \cdot 3 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 300d
consists of 2 curves linked by isogenies of
degree 2.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{5}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | 2.2.5.1-3600.1-d3 |
| $4$ | 4.0.8000.1 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.64000000.3 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.5184000000.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.44286750000.1 | \(\Z/6\Z\) | not in database |
| $16$ | 16.0.26873856000000000000.6 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | nonsplit | add | ord | ord | ss | ord | ss | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 1 | - | 1 | 1 | 1,1 | 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 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.