Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-31x-30\)
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(homogenize, simplify) |
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\(y^2z=x^3-31xz^2-30z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-31x-30\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-3, 6\right) \) | $0.92218023407559058483039719188$ | $\infty$ |
| \( \left(-1, 0\right) \) | $0$ | $2$ |
| \( \left(6, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-3:6:1]\) | $0.92218023407559058483039719188$ | $\infty$ |
| \([-1:0:1]\) | $0$ | $2$ |
| \([6:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-3, 6\right) \) | $0.92218023407559058483039719188$ | $\infty$ |
| \( \left(-1, 0\right) \) | $0$ | $2$ |
| \( \left(6, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-5, 0\right) \), \((-3,\pm 6)\), \( \left(-1, 0\right) \), \( \left(6, 0\right) \), \((13,\pm 42)\), \((17,\pm 66)\), \((699,\pm 18480)\)
\([-5:0:1]\), \([-3:\pm 6:1]\), \([-1:0:1]\), \([6:0:1]\), \([13:\pm 42:1]\), \([17:\pm 66:1]\), \([699:\pm 18480:1]\)
\( \left(-5, 0\right) \), \((-3,\pm 6)\), \( \left(-1, 0\right) \), \( \left(6, 0\right) \), \((13,\pm 42)\), \((17,\pm 66)\), \((699,\pm 18480)\)
Invariants
| Conductor: | $N$ | = | \( 616 \) | = | $2^{3} \cdot 7 \cdot 11$ |
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| Minimal Discriminant: | $\Delta$ | = | $1517824$ | = | $2^{8} \cdot 7^{2} \cdot 11^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{12869712}{5929} \) | = | $2^{4} \cdot 3^{3} \cdot 7^{-2} \cdot 11^{-2} \cdot 31^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.11919429049067985960227904201$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.58129241086397673254710045631$ |
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| $abc$ quality: | $Q$ | ≈ | $0.92154672345911$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.4119137518142453$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.92218023407559058483039719188$ |
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| Real period: | $\Omega$ | ≈ | $2.1143298453392748286765035623$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2^{2}\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.9497931916879797004327669941 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.949793192 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 2.114330 \cdot 0.922180 \cdot 16}{4^2} \\ & \approx 1.949793192\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 64 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{1}^{*}$ | additive | -1 | 3 | 8 | 0 |
| $7$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $11$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
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 |
|---|---|---|---|
| $2$ | 2Cs | 4.12.0.1 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 308 = 2^{2} \cdot 7 \cdot 11 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 45 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 155 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 57 & 4 \\ 114 & 9 \end{array}\right),\left(\begin{array}{rr} 305 & 4 \\ 304 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[308])$ is a degree-$53222400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/308\Z)$.
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 | $2$ | \( 1 \) |
| $7$ | split multiplicative | $8$ | \( 88 = 2^{3} \cdot 11 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 56 = 2^{3} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 616e
consists of 4 curves linked by isogenies of
degrees dividing 4.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{7}, \sqrt{11})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{11})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | \(\Q(i, \sqrt{7}, \sqrt{11})\) | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.932939628544.14 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.78724813483008.15 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | ss | ord | split | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 1,3 | 1 | 2 | 1 | 1 | 1 | 3 | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.