Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-48628x-3387192\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-48628xz^2-3387192z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-63021915x-157843764234\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-166, 426\right) \) | $1.8780463903026581211820776069$ | $\infty$ |
| \( \left(-\frac{321}{4}, \frac{321}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-166:426:1]\) | $1.8780463903026581211820776069$ | $\infty$ |
| \([-642:321:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-5973, 74088\right) \) | $1.8780463903026581211820776069$ | $\infty$ |
| \( \left(-2886, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-166, 426\right) \), \( \left(-166, -260\right) \), \( \left(262, 1238\right) \), \( \left(262, -1500\right) \), \( \left(2782, 144878\right) \), \( \left(2782, -147660\right) \)
\([-166:426:1]\), \([-166:-260:1]\), \([262:1238:1]\), \([262:-1500:1]\), \([2782:144878:1]\), \([2782:-147660:1]\)
\((-5973,\pm 74088)\), \((9435,\pm 295704)\), \((100155,\pm 31594104)\)
Invariants
| Conductor: | $N$ | = | \( 19166 \) | = | $2 \cdot 7 \cdot 37^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $2414841170339528$ | = | $2^{3} \cdot 7^{6} \cdot 37^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{4956477625}{941192} \) | = | $2^{-3} \cdot 5^{3} \cdot 7^{-6} \cdot 11^{3} \cdot 31^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.6692537924111418340672075719$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.13620516391097038811684026362$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0082122835525031$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.4610024623522815$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.8780463903026581211820776069$ |
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| Real period: | $\Omega$ | ≈ | $0.32573061108301131601708589796$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 36 $ = $ 3\cdot( 2 \cdot 3 )\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.5056347851997556642322042116 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.505634785 \approx L'(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.325731 \cdot 1.878046 \cdot 36}{2^2} \\ & \approx 5.505634785\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 103680 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $7$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $37$ | $2$ | $I_0^{*}$ | 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 | 8.6.0.6 | $6$ |
| $3$ | 3Cs | 3.12.0.1 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 18648 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 37 \), index $864$, genus $21$, and generators
$\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 1333 & 12876 \\ 16650 & 667 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 445 & 14652 \\ 11766 & 12877 \end{array}\right),\left(\begin{array}{rr} 17650 & 12099 \\ 15873 & 16984 \end{array}\right),\left(\begin{array}{rr} 19 & 24 \\ 1440 & 1819 \end{array}\right),\left(\begin{array}{rr} 18613 & 36 \\ 18612 & 37 \end{array}\right),\left(\begin{array}{rr} 9325 & 14652 \\ 0 & 1037 \end{array}\right),\left(\begin{array}{rr} 4031 & 0 \\ 0 & 18647 \end{array}\right)$.
The torsion field $K:=\Q(E[18648])$ is a degree-$25391279112192$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/18648\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 1369 = 37^{2} \) |
| $3$ | good | $2$ | \( 1369 = 37^{2} \) |
| $7$ | split multiplicative | $8$ | \( 2738 = 2 \cdot 37^{2} \) |
| $37$ | additive | $686$ | \( 14 = 2 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 19166a
consists of 6 curves linked by isogenies of
degrees dividing 18.
Twists
The minimal quadratic twist of this elliptic curve is 14a2, its twist by $37$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{37}) \) | \(\Z/6\Z\) | 2.2.37.1-196.1-h4 |
| $2$ | \(\Q(\sqrt{-111}) \) | \(\Z/6\Z\) | 2.0.111.1-196.5-f4 |
| $4$ | \(\Q(\sqrt{74 +74 \sqrt{-7}})\) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{37})\) | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{37})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{-111})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.4.385179247968256.7 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.294902861725696.72 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | \(\Q(\sqrt{2}, \sqrt{-3}, \sqrt{37})\) | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.0.4607857214464.5 | \(\Z/12\Z\) | not in database |
| $8$ | 8.0.373236434371584.30 | \(\Z/12\Z\) | 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 |
| $18$ | 18.6.696906606797966288843602962074253.1 | \(\Z/18\Z\) | not in database |
| $18$ | 18.0.56185703389611501417712177571870809288704.2 | \(\Z/18\Z\) | 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 | split | ord | ss | split | ss | ord | ord | ord | ss | ord | ord | add | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | 7 | 1,3 | 2 | 1,1 | 1 | 1 | 3 | 1,1 | 1 | 1 | - | 1 | 3 | 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.