Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2+7530x-418924\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z+7530xz^2-418924z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+9758205x-19691694594\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{68301164}{167281}, \frac{562256843642}{68417929}\right) \) | $17.119325030038517823032779072$ | $\infty$ |
| \( \left(44, -22\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([27935176076:562256843642:68417929]\) | $17.119325030038517823032779072$ | $\infty$ |
| \([44:-22:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{2461351119}{167281}, \frac{124464477242880}{68417929}\right) \) | $17.119325030038517823032779072$ | $\infty$ |
| \( \left(1599, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(44, -22\right) \)
\([44:-22:1]\)
\( \left(1599, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 23534 \) | = | $2 \cdot 7 \cdot 41^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-104274288298432$ | = | $-1 \cdot 2^{6} \cdot 7^{3} \cdot 41^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{9938375}{21952} \) | = | $2^{-6} \cdot 5^{3} \cdot 7^{-3} \cdot 43^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.3740072791612108591079253622$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.48277875419094304282545632432$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9869508090989833$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.9169648681151767$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $17.119325030038517823032779072$ |
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| Real period: | $\Omega$ | ≈ | $0.30943362686669440226652871389$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.2972948335546006637623160697 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.297294834 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.309434 \cdot 17.119325 \cdot 4}{2^2} \\ & \approx 5.297294834\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 69120 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $7$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $41$ | $2$ | $I_0^{*}$ | 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 |
|---|---|---|---|
| $2$ | 2B | 8.6.0.1 | $6$ |
| $3$ | 3Cs | 3.12.0.1 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 20664 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 41 \), index $864$, genus $21$, and generators
$\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 10333 & 19188 \\ 0 & 12629 \end{array}\right),\left(\begin{array}{rr} 10333 & 19188 \\ 9594 & 14761 \end{array}\right),\left(\begin{array}{rr} 7381 & 9348 \\ 9594 & 3691 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 11686 & 5043 \\ 19557 & 17836 \end{array}\right),\left(\begin{array}{rr} 19 & 24 \\ 1440 & 1819 \end{array}\right),\left(\begin{array}{rr} 20629 & 36 \\ 20628 & 37 \end{array}\right),\left(\begin{array}{rr} 1511 & 0 \\ 0 & 20663 \end{array}\right)$.
The torsion field $K:=\Q(E[20664])$ is a degree-$38392587878400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/20664\Z)$.
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 | $4$ | \( 11767 = 7 \cdot 41^{2} \) |
| $3$ | good | $2$ | \( 1681 = 41^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 3362 = 2 \cdot 41^{2} \) |
| $41$ | additive | $842$ | \( 14 = 2 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 23534e
consists of 6 curves linked by isogenies of
degrees dividing 18.
Twists
The minimal quadratic twist of this elliptic curve is 14a1, its twist by $41$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{41}) \) | \(\Z/6\Z\) | 2.2.41.1-196.1-h3 |
| $2$ | \(\Q(\sqrt{-123}) \) | \(\Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-41 -82 \sqrt{2}})\) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{41})\) | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-7}, \sqrt{41})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-7}, \sqrt{-123})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.0.340426706830336.6 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.27789935251456.56 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | \(\Q(\sqrt{-3}, \sqrt{-7}, \sqrt{41})\) | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.4.567141535744.4 | \(\Z/12\Z\) | not in database |
| $8$ | 8.0.45938464395264.12 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/3\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.6.1755552313974888466912116040275729.1 | \(\Z/18\Z\) | not in database |
| $18$ | 18.0.141535380258683820827570913364980424323072.1 | \(\Z/18\Z\) | 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 | ord | ss | nonsplit | ss | ord | ord | ord | ss | ord | ord | ord | add | ord | ord |
| $\lambda$-invariant(s) | 4 | 1 | 1,5 | 1 | 1,1 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | - | 1 | 1 |
| $\mu$-invariant(s) | 0 | 1 | 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.