Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-907x-9911\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-907xz^2-9911z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1176147x-444768786\)
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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 |
|---|---|---|
| \( \left(-20, 37\right) \) | $1.0842234349710148831632168441$ | $\infty$ |
| \( \left(-22, 11\right) \) | $0$ | $2$ |
| \( \left(34, -17\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-20:37:1]\) | $1.0842234349710148831632168441$ | $\infty$ |
| \([-22:11:1]\) | $0$ | $2$ |
| \([34:-17:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-705, 5832\right) \) | $1.0842234349710148831632168441$ | $\infty$ |
| \( \left(-777, 0\right) \) | $0$ | $2$ |
| \( \left(1239, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-22, 11\right) \), \( \left(-20, 37\right) \), \( \left(-20, -17\right) \), \( \left(-15, 32\right) \), \( \left(-15, -17\right) \), \( \left(34, -17\right) \), \( \left(48, 221\right) \), \( \left(48, -269\right) \), \( \left(59, 353\right) \), \( \left(59, -412\right) \), \( \left(223, 3196\right) \), \( \left(223, -3419\right) \)
\([-22:11:1]\), \([-20:37:1]\), \([-20:-17:1]\), \([-15:32:1]\), \([-15:-17:1]\), \([34:-17:1]\), \([48:221:1]\), \([48:-269:1]\), \([59:353:1]\), \([59:-412:1]\), \([223:3196:1]\), \([223:-3419:1]\)
\( \left(-777, 0\right) \), \((-705,\pm 5832)\), \((-525,\pm 5292)\), \( \left(1239, 0\right) \), \((1743,\pm 52920)\), \((2139,\pm 82620)\), \((8043,\pm 714420)\)
Invariants
| Conductor: | $N$ | = | \( 1470 \) | = | $2 \cdot 3 \cdot 5 \cdot 7^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $8576612100$ | = | $2^{2} \cdot 3^{6} \cdot 5^{2} \cdot 7^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{702595369}{72900} \) | = | $2^{-2} \cdot 3^{-6} \cdot 5^{-2} \cdot 7^{3} \cdot 127^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.64124132512998083243568001409$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.33171374939767582011699635763$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0045688963827404$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.394031936841815$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.0842234349710148831632168441$ |
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| Real period: | $\Omega$ | ≈ | $0.87566212778174627644420521041$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.8988268001151054181229914035 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.898826800 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.875662 \cdot 1.084223 \cdot 32}{4^2} \\ & \approx 1.898826800\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1152 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $3$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $5$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $7$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 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 |
|---|---|---|---|
| $2$ | 2Cs | 2.6.0.1 | $6$ |
| $3$ | 3B | 3.4.0.1 | $4$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 631 & 252 \\ 546 & 673 \end{array}\right),\left(\begin{array}{rr} 337 & 126 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 561 & 364 \\ 742 & 729 \end{array}\right),\left(\begin{array}{rr} 829 & 12 \\ 828 & 13 \end{array}\right),\left(\begin{array}{rr} 9 & 4 \\ 824 & 833 \end{array}\right),\left(\begin{array}{rr} 599 & 0 \\ 0 & 839 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 547 & 126 \\ 714 & 715 \end{array}\right)$.
The torsion field $K:=\Q(E[840])$ is a degree-$185794560$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/840\Z)$.
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 | $4$ | \( 49 = 7^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 490 = 2 \cdot 5 \cdot 7^{2} \) |
| $5$ | split multiplicative | $6$ | \( 294 = 2 \cdot 3 \cdot 7^{2} \) |
| $7$ | additive | $26$ | \( 30 = 2 \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 1470.d
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 30.a6, its twist by $-7$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | 2.0.7.1-900.2-a5 |
| $4$ | \(\Q(\sqrt{6}, \sqrt{14})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{10}, \sqrt{-14})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-15}, \sqrt{-21})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $6$ | 6.2.92610000.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $8$ | \(\Q(\sqrt{-2}, \sqrt{-3}, \sqrt{-7})\) | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | \(\Q(\sqrt{2}, \sqrt{5}, \sqrt{-7})\) | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | \(\Q(\sqrt{3}, \sqrt{-5}, \sqrt{-7})\) | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $12$ | 12.0.8576612100000000.1 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | \(\Q(i, \sqrt{6}, \sqrt{10}, \sqrt{14})\) | \(\Z/4\Z \oplus \Z/4\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $18$ | 18.0.830844982933255254894300000000.1 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | nonsplit | split | add | ss | ord | ord | ord | ss | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 4 | 1 | 2 | - | 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 | 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.