Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-86098965x-307528130376\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-86098965xz^2-307528130376z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-6974016192x-224167084995555\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(21012, 2675946)$ | $3.7496631612893518016427138698$ | $\infty$ |
| $(-5352, 0)$ | $0$ | $2$ |
Integral points
\( \left(-5352, 0\right) \), \((21012,\pm 2675946)\)
Invariants
| Conductor: | $N$ | = | \( 14196 \) | = | $2^{2} \cdot 3 \cdot 7 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $130968975797156467152$ | = | $2^{4} \cdot 3^{8} \cdot 7^{6} \cdot 13^{9} $ |
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| j-invariant: | $j$ | = | \( \frac{416013434950254592}{771895089} \) | = | $2^{26} \cdot 3^{-8} \cdot 7^{-6} \cdot 11^{3} \cdot 167^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.1161260093651904048727822243$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.96136493108238941636025593597$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1844185766908968$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.94787192380638$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.7496631612893518016427138698$ |
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| Real period: | $\Omega$ | ≈ | $0.049562147962982027170478867260$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 96 $ = $ 1\cdot2^{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)$ | ≈ | $4.4601926498679790990228148664 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.460192650 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.049562 \cdot 3.749663 \cdot 96}{2^2} \\ & \approx 4.460192650\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1797120 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $3$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $7$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $13$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 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 |
|---|---|---|
| $2$ | 2B | 8.12.0.31 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \), index $96$, genus $3$, and generators
$\left(\begin{array}{rr} 11 & 16 \\ 4304 & 4275 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 13 & 8 \\ 8 & 5 \end{array}\right),\left(\begin{array}{rr} 4353 & 16 \\ 4352 & 17 \end{array}\right),\left(\begin{array}{rr} 1696 & 5 \\ 2403 & 16 \end{array}\right),\left(\begin{array}{rr} 1877 & 26 \\ 3014 & 3817 \end{array}\right),\left(\begin{array}{rr} 3 & 28 \\ 4292 & 3659 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 3831 & 10 \\ 1010 & 4277 \end{array}\right),\left(\begin{array}{rr} 1469 & 16 \\ 264 & 325 \end{array}\right),\left(\begin{array}{rr} 1105 & 16 \\ 264 & 325 \end{array}\right)$.
The torsion field $K:=\Q(E[4368])$ is a degree-$649240510464$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4368\Z)$.
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 | $2$ | \( 13 \) |
| $3$ | split multiplicative | $4$ | \( 676 = 2^{2} \cdot 13^{2} \) |
| $7$ | split multiplicative | $8$ | \( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \) |
| $13$ | additive | $62$ | \( 84 = 2^{2} \cdot 3 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 14196q
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 14196l1, its twist by $13$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{13}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.140608.1 | \(\Z/4\Z\) | not in database |
| $8$ | 8.4.47469233803264.11 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.19770609664.3 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.13680875742768.1 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | 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 | split | ord | split | ss | add | ord | ss | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 2 | 1 | 2 | 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 |
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.